Shortest known proofs of the propositional calculus theorems from ----------------------------------------------------------------- _Principia Mathematica_ ----------------------- ------------------------------------------------------------------------ PUBLIC DOMAIN This file was released into the Public Domain per the Creative Commons Public Domain Dedication by Norman Megill and Gregory Bush, on Feb. 16, 2007. See http://creativecommons.org/licenses/publicdomain/ The public domain release applies worldwide. In case this is not legally possible, the right is granted to use the work for any purpose, without any conditions, unless such conditions are required by law. ------------------------------------------------------------------------ Revision history 4-May-2004: Gregory Bush found shorter proofs for 67 of the original 193 proofs. This file has been updated with them. When you see a comment like "(GB 35->25)" after the theorem, it means the original proof had 35 steps and Greg found a shorter version with only 25 steps. 6-Mar-2008: Corrected theorems *2.36, *2.37, and *2.38, which were shown with v expanded into ~ and ->. (The "Result of proof" and the proof itself were still correct, however). 12-Jun-2010: A proof of Meredith's single axiom for propositional calculus was added. Contributed by Scott Fenton. ------------------------------------------------------------------------ This file contains proofs of all 193 theorems of propositional calculus in Whitehead and Russell's _Principia Mathematica_, directly from axioms ax-1, ax-2, ax-3, and ax-mp in the Metamath Solitaire applet (http://metamath.org). These are the shortest known direct proofs. If you find a shorter one, let me know at nm (at) alum (dot) mit (dot) edu (include the word "Metamath" in the subject). In these proofs, the resulting theorem is in a form which has all abbreviations expanded, so instead of "(~ P v P)" for theorem *2.1 you will see "(~ ~ P -> P)" or in Metamath Solitaire's ASCII notation, "(-. -. P -> P)". To get "(~ P v P)" we would apply the definition of logical OR, which is shown in the applet using: Select Logic Family -> Propositional Calculus + Definitions -> Axiom Information -> Axioms -> df-or. If we wanted we could extend the proof to apply the definition directly in the proof, but that is somewhat tedious and was not our purpose here. In addition, sometimes "more general" theorems are proved. For example, the proof of theorem *1.6 actually proves "((P -> Q) -> ((R -> P) -> (R -> Q)))" in the applet. We obtain theorem *1.6 by substituting "-. R" for "R", then applying the definition of logical OR. The notation for proofs is as follows. For theorem *1.6, for example, the proof shown is "DD2D121". To enter this into the applet, you enter the steps in _reverse_ order i.e. "121D2DD". The character mapping is as follows: "1" is ax-1, "2" is ax-2, "3" is ax-3, and "D" is ax-mp. Thus this proof would be entered into the applet as: ax-1 ax-2 ax-1 ax-mp ax-2 ax-mp ax-mp As a historical note, the notation in "DD2D121" is called "condensed detachment" and expresses the proof in Polish notation. It was invented by logician C. A. Meredith in the 1950's. For more information on this notation you can reference the article "A Finitely Axiomatized Formalization of Predicate Calculus with Equality" linked to from the Metamath Home Page. The Appendix in that article shows how to write a very simple program for verifying these proofs. For longer proofs such a program is much more convenient than repeated clicks on the applet. If this interests you, I have a hastily-written C program, drule.c, that implements the algorithm and which was used to verify these proofs. It is unpolished and probably not suitable for general release, but it has no known bugs. Download here: http://us.metamath.org/downloads/drule.c To compile: gcc drule.c -o drule To run: ./drule < pmproofs.txt > newpmproofs.txt where "pmproofs.txt" is the name of this file. Then drule.c will verify each proof below and also regenerate the "Result of proof" and the number of steps in the output file "newpmproofs.txt". See comments at the top of drule.c for more information. Note: everything up to and including the LAST LINE OF DASHES below is treated as a comment, i.e. discarded, by the drule.c proof checker. In addition, due to the crudeness of drule.c, the text above the last line of dashes MUST NOT CONTAIN SEMICOLONS. This is not a bug but is an intentional design shortcut made to write the program quickly. If you want to enhance it, be my guest. :) ------------------------------------------------------------------------ For each theorem, we list: (1) the theorem as it appears in _Principia Mathematica_ along with P.M.'s theorem number, (2) the unabbreviated result of the proof as it will appear in the Metamath Solitaire Java applet, and (3) the proof in the condensed detachment notation explained above. A semicolon ends the theorem or proof, and an exclamation point means the rest of the line is a comment. In the condensed detachment notation, "D" stands for the second of its two arguments detached from the first (i.e. the first argument is the major premise, and the second the minor premise, of modus ponens). "1", "2", and "3" stand for the three axioms of the system called P_2 (attributed to Lukasiewicz and popularized by Church): ax-1 (P -> (Q -> P)) ax-2 ((P -> (Q -> R)) -> ((P -> Q) -> (P -> R))) ax-3 ((~ P -> ~ Q) -> (Q -> P)) ------------------------------------------------------------------------- For drule.c: THERE MUST BE NO SEMICOLONS IN THE TEXT ABOVE For drule.c: THE COMMENT HEADER ABOVE ENDS WITH THE LINE OF DASHES BELOW ------------------------------------------------------------------------- ((P v P) -> P); ! *1.2 Taut ((~ P -> P) -> P); ! Result of proof DD2DD2D13D2DD2D1311; ! 19 steps (Q -> (P v Q)); ! *1.3 Add (P -> (Q -> P)); ! Result of proof 1; ! 1 step ((P v Q) -> (Q v P)); ! *1.4 ((~ P -> Q) -> (~ Q -> P)); ! Result of proof DD2D13D2D1D3DD2DD2D13DD2D1311; ! 29 steps ((P v (Q v R)) -> (Q v (P v R))); ! *1.5 Assoc ((P -> (Q -> R)) -> (Q -> (P -> R))); ! Result of proof DD2D1DD22D11DD2D112; ! 19 steps ((Q -> R) -> ((P v Q) -> (P v R))); ! *1.6 Sum ((P -> Q) -> ((R -> P) -> (R -> Q))); ! Result of proof DD2D121; ! 7 steps ((P -> ~ P) -> ~ P); ! *2.01 ((P -> ~ P) -> ~ P); ! Result of proof DD2D1DD2DD2D13D2DD2D1311DD2D1DD22D2DD2D13DD2D1311; ! 49 steps (Q -> (P -> Q)); ! *2.02 Simp (P -> (Q -> P)); ! Result of proof 1; ! 1 step ((P -> ~ Q) -> (Q -> ~ P)); ! *2.03 ((P -> ~ Q) -> (Q -> ~ P)); ! Result of proof DD2D13DD2D1DD22D2DD2D13DD2D1311; ! 31 steps ((P -> (Q -> R)) -> (Q -> (P -> R))); ! *2.04 Comm ((P -> (Q -> R)) -> (Q -> (P -> R))); ! Result of proof DD2D1DD22D11DD2D112; ! 19 steps ((Q -> R) -> ((P -> Q) -> (P -> R))); ! *2.05 Syll ((P -> Q) -> ((R -> P) -> (R -> Q))); ! Result of proof DD2D121; ! 7 steps ((P -> Q) -> ((Q -> R) -> (P -> R))); ! *2.06 Syll ((P -> Q) -> ((Q -> R) -> (P -> R))); ! Result of proof DD2D1D2DD2D1211; ! 15 steps (P -> (P v P)); ! *2.07 (P -> (Q -> P)); ! Result of proof 1; ! 1 step (P -> P); ! *2.08 Id (P -> P); ! Result of proof DD211; ! 5 steps (~ P v P); ! *2.1 (~ ~ P -> P); ! Result of proof DD2DD2D13DD2D1311; ! 17 steps (P v ~ P); ! *2.11 (P -> P); ! Result of proof DD211; ! 5 steps (P -> ~ ~ P); ! *2.12 (P -> ~ ~ P); ! Result of proof D3DD2DD2D13DD2D1311; ! 19 steps (P v ~ ~ ~ P); ! *2.13 (P -> ~ ~ P); ! Result of proof D3DD2DD2D13DD2D1311; ! 19 steps (~ ~ P -> P); ! *2.14 (~ ~ P -> P); ! Result of proof DD2DD2D13DD2D1311; ! 17 steps ((~ P -> Q) -> (~ Q -> P)); ! *2.15 Transp ((~ P -> Q) -> (~ Q -> P)); ! Result of proof DD2D13D2D1D3DD2DD2D13DD2D1311; ! 29 steps ((P -> Q) -> (~ Q -> ~ P)); ! *2.16 ((P -> Q) -> (~ Q -> ~ P)); ! Result of proof DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311 ; ! 59 steps ((~ Q -> ~ P) -> (P -> Q)); ! *2.17 Transp ((~ P -> ~ Q) -> (Q -> P)); ! Result of proof 3; ! 1 step ((~ P -> P) -> P); ! *2.18 ((~ P -> P) -> P); ! Result of proof DD2DD2D13D2DD2D1311; ! 19 steps (P -> (P v Q)); ! *2.2 (P -> (~ P -> Q)); ! Result of proof DD2D1D2DD2D1311; ! 15 steps (~ P -> (P -> Q)); ! *2.21 (~ P -> (P -> Q)); ! Result of proof DD2D131; ! 7 steps (P -> (~ P -> Q)); ! *2.24 (P -> (~ P -> Q)); ! Result of proof DD2D1D2DD2D1311; ! 15 steps (P v ((P v Q) -> Q)); ! *2.25 (P -> ((P -> Q) -> Q)); ! Result of proof DD2D1D2DD2111; ! 13 steps (~ P v ((P -> Q) -> Q)); ! *2.26 (GB 35->25) (~ ~ P -> ((P -> Q) -> Q)); ! Result of proof DD2D1D2DD211DD2D13DD2D131; ! 25 steps (P -> ((P -> Q) -> Q)); ! *2.27 (P -> ((P -> Q) -> Q)); ! Result of proof DD2D1D2DD2111; ! 13 steps ((P v (Q v R)) -> (P v (R v Q))); ! *2.3 ((P -> (~ Q -> R)) -> (P -> (~ R -> Q))); ! Result of proof D2D1DD2D13D2D1D3DD2DD2D13DD2D1311; ! 33 steps ((P v (Q v R)) -> ((P v Q) v R)); ! *2.31 ((P -> (~ Q -> R)) -> (~ (P -> Q) -> R)); ! Result of proof DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112D2D1DD2D13D 2D1D3DD2DD2D13DD2D1311; ! 91 steps (((P v Q) v R) -> (P v (Q v R))); ! *2.32 ((~ (P -> Q) -> R) -> (P -> (~ Q -> R))); ! Result of proof DD2D1D2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112DD2D13D 2D1D3DD2DD2D13DD2D1311; ! 91 steps ((Q -> R) -> ((P v Q) -> (R v P))); ! *2.36 ((P -> Q) -> ((~ R -> P) -> (~ Q -> R))); ! Result of proof DD2D1D2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D121; ! 45 steps ((Q -> R) -> ((Q v P) -> (P v R))); ! *2.37 ((P -> Q) -> ((~ P -> R) -> (~ R -> Q))); ! Result of proof DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D121; ! 53 steps ((Q -> R) -> ((Q v P) -> (R v P))); ! *2.38 ((P -> Q) -> ((~ P -> R) -> (~ Q -> R))); ! Result of proof DD2D1DD2D1D2DD2D1211DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D 13DD2D1311; ! 79 steps ((P v (P v Q)) -> (P v Q)); ! *2.4 ((P -> (P -> Q)) -> (P -> Q)); ! Result of proof DD22D21; ! 7 steps ((Q v (P v Q)) -> (P v Q)); ! *2.41 ((~ P -> (Q -> P)) -> (Q -> P)); ! Result of proof DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD22D11DD2D112; ! 47 steps ((~ P v (P -> Q)) -> (P -> Q)); ! *2.42 ((~ ~ P -> (P -> Q)) -> (P -> Q)); ! Result of proof DD2D1DD22D21DD2DD2D121D1D3DD2DD2D13DD2D1311; ! 43 steps ((P -> (P -> Q)) -> (P -> Q)); ! *2.43 ((P -> (P -> Q)) -> (P -> Q)); ! Result of proof DD22D21; ! 7 steps (~ (P v Q) -> ~ P); ! *2.45 (~ (P -> Q) -> P); ! Result of proof D3DD2D1D3DD2DD2D13DD2D1311DD2D131; ! 33 steps (~ (P v Q) -> ~ Q); ! *2.46 (~ (P -> Q) -> ~ Q); ! Result of proof D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D131; ! 39 steps (~ (P v Q) -> (~ P v Q)); ! *2.47 (GB 35->31) (~ (P -> Q) -> (~ P -> R)); ! Result of proof DD2D1DD22D1DD2D131DD2D11DD2D131; ! 31 steps (~ (P v Q) -> (P v ~ Q)); ! *2.48 (GB 45->43) (~ (P -> Q) -> (R -> ~ Q)); ! Result of proof DD2D1DD2D1DD2D13DD22D1DD2D13DD2D1311DD2D131; ! 43 steps (~ (P v Q) -> (~ P v ~ Q)); ! *2.49 (GB 35->31) (~ (P -> Q) -> (~ P -> R)); ! Result of proof DD2D1DD22D1DD2D131DD2D11DD2D131; ! 31 steps (~ (P -> Q) -> (~ P -> Q)); ! *2.5 (GB 35->31) (~ (P -> Q) -> (~ P -> R)); ! Result of proof DD2D1DD22D1DD2D131DD2D11DD2D131; ! 31 steps (~ (P -> Q) -> (P -> ~ Q)); ! *2.51 (GB 45->43) (~ (P -> Q) -> (R -> ~ Q)); ! Result of proof DD2D1DD2D1DD2D13DD22D1DD2D13DD2D1311DD2D131; ! 43 steps (~ (P -> Q) -> (~ P -> ~ Q)); ! *2.52 (GB 35->31) (~ (P -> Q) -> (~ P -> R)); ! Result of proof DD2D1DD22D1DD2D131DD2D11DD2D131; ! 31 steps (~ (P -> Q) -> (Q -> P)); ! *2.521 (GB 29->25) (~ (P -> Q) -> (Q -> R)); ! Result of proof DD2D1DD22D11DD2D11DD2D131; ! 25 steps ((P v Q) -> (~ P -> Q)); ! *2.53 (P -> P); ! Result of proof DD211; ! 5 steps ((~ P -> Q) -> (P v Q)); ! *2.54 (P -> P); ! Result of proof DD211; ! 5 steps (~ P -> ((P v Q) -> Q)); ! *2.55 (P -> ((P -> Q) -> Q)); ! Result of proof DD2D1D2DD2111; ! 13 steps (~ Q -> ((P v Q) -> P)); ! *2.56 (GB 37->35) (~ P -> ((~ Q -> P) -> Q)); ! Result of proof DD2D1D2DD2D1D231DD2D12DD2D11DD2D131; ! 35 steps ((~ P -> Q) -> ((P -> Q) -> Q)); ! *2.6 ((~ P -> Q) -> ((P -> Q) -> Q)); ! Result of proof DD2D1D2DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD 2D13DD2D1311DD2D1211; ! 89 steps ((P -> Q) -> ((~ P -> Q) -> Q)); ! *2.61 ((P -> Q) -> ((~ P -> Q) -> Q)); ! Result of proof DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2 D1311DD2D121; ! 81 steps ((P v Q) -> ((P -> Q) -> Q)); ! *2.62 ((~ P -> Q) -> ((P -> Q) -> Q)); ! Result of proof DD2D1D2DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD 2D13DD2D1311DD2D1211; ! 89 steps ((P -> Q) -> ((P v Q) -> Q)); ! *2.621 ((P -> Q) -> ((~ P -> Q) -> Q)); ! Result of proof DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2 D1311DD2D121; ! 81 steps ((P v Q) -> ((~ P v Q) -> Q)); ! *2.63 ((P -> Q) -> ((~ P -> Q) -> Q)); ! Result of proof DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2 D1311DD2D121; ! 81 steps ((P v Q) -> ((P v ~ Q) -> P)); ! *2.64 (GB 61->39) ((~ P -> Q) -> ((~ P -> ~ Q) -> P)); ! Result of proof DD2D13D2DD2D13DD2D1D2DD2DD2D13DD2D13111; ! 39 steps ((P -> Q) -> ((P -> ~ Q) -> ~ P)); ! *2.65 ((P -> Q) -> ((P -> ~ Q) -> ~ P)); ! Result of proof DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2DD2D13DD2D1D2DD2DD2D13DD2D13111 ; ! 69 steps (((P v Q) -> Q) -> (P -> Q)); ! *2.67 (((~ P -> Q) -> R) -> (P -> R)); ! Result of proof DD2DD2D121D1DD2D1D2DD2D1311; ! 27 steps (((P -> Q) -> Q) -> (P v Q)); ! *2.68 (((P -> Q) -> R) -> (~ P -> R)); ! Result of proof DD2DD2D121D1DD2D131; ! 19 steps (((P -> Q) -> Q) -> ((Q -> P) -> P)); ! *2.69 (((P -> Q) -> R) -> ((R -> P) -> P)); ! Result of proof DD2D1DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1D2DD2D1211DD2DD2D121D1DD2D131 ; ! 67 steps ((P -> Q) -> (((P v Q) v R) -> (Q v R))); ! *2.73 ((P -> Q) -> ((~ (~ P -> Q) -> R) -> (~ Q -> R))); ! Result of proof DD2D1DD2D1D2DD2D1211DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD 2DD2D13DD2D1311DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D 1D3DD2DD2D13DD2D1311DD2D121; ! 165 steps ((Q -> P) -> (((P v Q) v R) -> (P v R))); ! *2.74 ((P -> Q) -> ((~ (~ Q -> P) -> R) -> (~ Q -> R))); ! Result of proof DD2D1DD2D1D2DD2D1211DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD 2DD2D13DD2D1311DD2D1D2D1DD2DD2D13D2DD2D1311DD2D121; ! 119 steps ((P v Q) -> ((P v (Q -> R)) -> (P v R))); ! *2.75 ((P -> Q) -> ((P -> (Q -> R)) -> (P -> R))); ! Result of proof DD2D1D221; ! 9 steps ((P v (Q -> R)) -> ((P v Q) -> (P v R))); ! *2.76 ((P -> (Q -> R)) -> ((P -> Q) -> (P -> R))); ! Result of proof 2; ! 1 step ((P -> (Q -> R)) -> ((P -> Q) -> (P -> R))); ! *2.77 ((P -> (Q -> R)) -> ((P -> Q) -> (P -> R))); ! Result of proof 2; ! 1 step ((Q v R) -> ((~ R v S) -> (Q v S))); ! *2.8 (GB 63->43) ((P -> Q) -> ((~ ~ Q -> R) -> (P -> R))); ! Result of proof DD2D1DD2D1D2DD2D1211D2D1D3DD2DD2D13DD2D1311; ! 43 steps ((Q -> (R -> S)) -> ((P v Q) -> ((P v R) -> (P v S)))); ! *2.81 ((P -> (Q -> R)) -> ((S -> P) -> ((S -> Q) -> (S -> R)))) ; ! Result of proof DD2D1D2D12DD2D121; ! 17 steps (((P v Q) v R) -> (((P v ~ R) v S) -> ((P v Q) v S))); ! *2.82 ((~ (P -> Q) -> R) -> ((~ (P -> ~ R) -> S) -> (~ (P -> Q) -> S))) ; ! Result of proof DD2D1DD2D1D2DD2D1211D2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111D3DD2D1D3DD 2DD2D13DD2D1311DD2D131; ! 91 steps ((P -> (Q -> R)) -> ((P -> (R -> S)) -> (P -> (Q -> S)))); ! *2.83 ((P -> (Q -> R)) -> ((P -> (R -> S)) -> (P -> (Q -> S)))) ; ! Result of proof DD2D12D2D1DD2D1D2DD2D1211; ! 25 steps (((P v Q) -> (P v R)) -> (P v (Q -> R))); ! *2.85 (GB 41->37) (((P -> Q) -> (R -> S)) -> (R -> (Q -> S))); ! Result of proof DD2D1DD2DD2D121D11DD2D12DD2DD2D121D11; ! 37 steps (((P -> Q) -> (P -> R)) -> (P -> (Q -> R))); ! *2.86 (GB 41->37) (((P -> Q) -> (R -> S)) -> (R -> (Q -> S))); ! Result of proof DD2D1DD2DD2D121D11DD2D12DD2DD2D121D11; ! 37 steps ((P ^ Q) -> ~ (~ P v ~ Q)); ! *3.1 (GB 79->73) (~ (P -> Q) -> ~ (~ ~ P -> Q)); ! Result of proof D3DD2D1D3DD2DD2D13DD2D1311DDD22D1D1D3DD2DD2D13DD2D1311DD2D12DD2D13DD2 D131; ! 73 steps (~ (~ P v ~ Q) -> (P ^ Q)); ! *3.11 (GB 73->63) (~ (~ ~ P -> Q) -> ~ (P -> Q)); ! Result of proof D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD22D2DD2D13DD2D131DD2D13DD2D131 ; ! 63 steps ((~ P v ~ Q) v (P ^ Q)); ! *3.12 (GB 73->63) (~ (~ ~ P -> Q) -> ~ (P -> Q)); ! Result of proof D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD22D2DD2D13DD2D131DD2D13DD2D131 ; ! 63 steps (~ (P ^ Q) -> (~ P v ~ Q)); ! *3.13 (GB 47->37) (~ ~ (P -> Q) -> (~ ~ P -> Q)); ! Result of proof DD2D1DD22D2DD2D13DD2D131DD2D13DD2D131; ! 37 steps ((~ P v ~ Q) -> ~ (P ^ Q)); ! *3.14 ((~ ~ P -> Q) -> ~ ~ (P -> Q)); ! Result of proof DD2D1D3DD2DD2D13DD2D1311DD2DD2D121D1D3DD2DD2D13DD2D1311; ! 55 steps (P -> (Q -> (P ^ Q))); ! *3.2 (P -> (Q -> ~ (P -> ~ Q))); ! Result of proof DD2D13DD2D1D2DD2DD2D13DD2D13111; ! 31 steps (Q -> (P -> (P ^ Q))); ! *3.21 (P -> (Q -> ~ (Q -> ~ P))); ! Result of proof DD2D1D2DD2D13DD2D1D2DD2DD2D13DD2D131111; ! 39 steps ((P ^ Q) -> (Q ^ P)); ! *3.22 (GB 79->69) (~ (P -> ~ Q) -> ~ (Q -> ~ P)); ! Result of proof D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D1DD22D2DD2D13DD2D131DD2D13DD2D131 ; ! 69 steps ~ (P ^ ~ P); ! *3.24 ~ ~ (P -> ~ ~ P); ! Result of proof DD3DD2DD2D13DD2D1311D3DD2DD2D13DD2D1311; ! 39 steps ((P ^ Q) -> P); ! *3.26 Simp (~ (P -> Q) -> P); ! Result of proof D3DD2D1D3DD2DD2D13DD2D1311DD2D131; ! 33 steps ((P ^ Q) -> Q); ! *3.27 Simp (~ (P -> ~ Q) -> Q); ! Result of proof D3DD2D1D3DD2DD2D13DD2D13111; ! 27 steps (((P ^ Q) -> R) -> (P -> (Q -> R))); ! *3.3 Exp (GB 63->59) ((~ (P -> ~ Q) -> R) -> (P -> (Q -> R))); ! Result of proof DD2D1D2DD2D1DD2D1D2DD2D1211DD2D13DD2D1D2DD2DD2D13DD2D131111 ; ! 59 steps ((P -> (Q -> R)) -> ((P ^ Q) -> R)); ! *3.31 Imp ((P -> (Q -> R)) -> (~ (P -> ~ Q) -> R)); ! Result of proof DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112D2D1DD2D1DD 2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311; ! 121 steps (((P -> Q) ^ (Q -> R)) -> (P -> R)); ! *3.33 Syll (GB 113->105) (~ ((P -> Q) -> ~ (Q -> R)) -> (P -> R)); ! Result of proof D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D1DD2D1DD2D1DD2133D2D1D3DD2DD2D13 DD2D1311DD2D1D2DD2D12DD2D13DD2D13111; ! 105 steps (((Q -> R) ^ (P -> Q)) -> (P -> R)); ! *3.34 Syll (~ ((P -> Q) -> ~ (R -> P)) -> (R -> Q)); ! Result of proof D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2 D1311D2D1D3DD2DD2D13DD2D1311DD2D1211; ! 105 steps ((P ^ (P -> Q)) -> Q); ! *3.35 Ass (GB 93->63) (~ (P -> ~ (P -> Q)) -> Q); ! Result of proof DD2D3DD2D1D3DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2D1311DD2D131 ; ! 63 steps (((P ^ Q) -> R) -> ((P ^ ~ R) -> ~ Q)); ! *3.37 Transp ((~ (P -> ~ Q) -> R) -> (~ (P -> ~ ~ R) -> ~ Q)); ! Result of proof DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1 DD2D1DD2D1DD2D1DD22D11DD2D112D2D1DD2D1D2D1D3DD2DD2D13DD2D13113DD2D1DD 22D11DD2D112DD2D13D2D1D3DD2DD2D13DD2D1311; ! 179 steps ((P ^ Q) -> (P -> Q)); ! *3.4 (GB 33->31) (~ (P -> ~ Q) -> (R -> Q)); ! Result of proof DD2D1DD2D13DD22D11DD2D11DD2D131; ! 31 steps ((P -> R) -> ((P ^ Q) -> R)); ! *3.41 ((P -> Q) -> (~ (P -> R) -> Q)); ! Result of proof DD2DD2D121D1D3DD2D1D3DD2DD2D13DD2D1311DD2D131; ! 45 steps ((Q -> R) -> ((P ^ Q) -> R)); ! *3.42 ((P -> Q) -> (~ (R -> ~ P) -> Q)); ! Result of proof DD2DD2D121D1D3DD2D1D3DD2DD2D13DD2D13111; ! 39 steps (((P -> Q) ^ (P -> R)) -> (P -> (Q ^ R))); ! *3.43 Comp (~ ((P -> Q) -> ~ (P -> R)) -> (P -> ~ (Q -> ~ R))) ; ! Result of proof D3DD2D1D3DD2DD2D13DD2D1311DDD22D11D1D2DD2D1DD2D13D2D1D3DD2DD2D13DD2D1 311DD2D1DD2D1DD22D2DD2D13DD2D1311DD2D12D2D1DD2D13DD2D1D2DD2DD2D13DD2D 13111; ! 143 steps (((Q -> P) ^ (R -> P)) -> ((Q v R) -> P)); ! *3.44 (~ ((P -> Q) -> ~ (R -> Q)) -> ((~ P -> R) -> Q)); ! Result of proof DD2D13DD2DD2D1DD2D12D2D1DD2D13DD2D1D2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D D2D13DD2D13111DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D1 3DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD2D1DD2D1DD2D13DD2D1DD22D2 DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D13111 ; ! 271 steps ((P -> Q) -> ((P ^ R) -> (Q ^ R))); ! *3.45 Fact (GB 79->71) ((P -> Q) -> (~ (P -> R) -> ~ (Q -> R))); ! Result of proof DD2D1DD2D1DD2D1DD2133D2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D12DD2D13DD2D13 11; ! 71 steps (((P -> R) ^ (Q -> S)) -> ((P ^ Q) -> (R ^ S))); ! *3.47 (~ ((P -> Q) -> ~ (R -> S)) -> (~ (P -> ~ R) -> ~ (Q -> ~ S))) ; ! Result of proof DD2DD2D1DD2D12D2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1DD2DD2D121D1D3D D2D1D3DD2DD2D13DD2D1311DD2D131D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD2D1D D2DD2D121D1D3DD2D1D3DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2D13111 ; ! 203 steps (((P -> R) ^ (Q -> S)) -> ((P v Q) -> (R v S))) ; ! *3.48 (GB 245->241) (~ ((P -> Q) -> ~ (R -> S)) -> ((~ P -> R) -> (~ Q -> S))) ; ! Result of proof DD2D1D2DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112DD2D 1D2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1 DD2DD2D121D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D13 11DD2D1DD2D1D2D1DD2D1D2DD2D1211211; ! 241 steps ((P -> Q) <-> (~ Q -> ~ P)); ! *4.1 ~ (((P -> Q) -> (~ Q -> ~ P)) -> ~ ((~ R -> ~ S) -> (S -> R))) ; ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD 2DD2D13DD2D13113; ! 85 steps ((P <-> Q) <-> (~ P <-> ~ Q)); ! *4.11 ~ ((~ ((P -> Q) -> ~ (R -> S)) -> ~ ((~ S -> ~ R) -> ~ (~ Q -> ~ P))) -> ~ (~ ((~ T -> ~ U) -> ~ (~ V -> ~ W)) -> ~ ((W -> V) -> ~ (U -> T)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1 DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311D3DD2D1D3D D2DD2D13DD2D13111DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD 2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD2DD2D1DD2D13DD2D1D2DD2 DD2D13DD2D13111DD2D13D3DD2D1D3DD2DD2D13DD2D13111DD2D13D3DD2D1D3DD2DD2 D13DD2D1311DD2D131; ! 363 steps ((P <-> ~ Q) <-> (Q <-> ~ P)); ! *4.12 ~ ((~ ((P -> ~ Q) -> ~ (~ R -> S)) -> ~ ((Q -> ~ P) -> ~ (~ S -> R))) -> ~ (~ ((T -> ~ U) -> ~ (~ V -> W)) -> ~ ((U -> ~ T) -> ~ (~ W -> V)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1 DD2D13DD2D1DD22D2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD2D1 DD2D13D2D1D3DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D13111DD2DD2D1DD2D1 3DD2D1D2DD2DD2D13DD2D13111DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D3DD2D1 D3DD2DD2D13DD2D1311DD2D131DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311D3DD2D1D3 DD2DD2D13DD2D13111; ! 363 steps (P <-> ~ ~ P); ! *4.13 ~ ((P -> ~ ~ P) -> ~ (~ ~ Q -> Q)); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2DD2D13DD2D1311DD2DD2D13DD2D1311 ; ! 61 steps (((P ^ Q) -> R) <-> ((P ^ ~ R) -> ~ Q)); ! *4.14 (GB 325->321) ~ (((~ (P -> ~ Q) -> R) -> (~ (P -> ~ ~ R) -> ~ Q)) -> ~ ((~ (S -> ~ ~ T) -> U) -> (~ (S -> U) -> T))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D 1D3DD2DD2D13DD2D1311DD2D1DD2D1DD2D1DD2D1DD22D11DD2D112D2D1DD2D1D2D1D3 DD2DD2D13DD2D13113DD2D1DD22D11DD2D112DD2D13D2D1D3DD2DD2D13DD2D1311DD2 D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112DD2D1D2DD2D1DD 2D1D2DD2D1211DD2D13DD2D1D2DD2DD2D13DD2D131111; ! 321 steps (((P ^ Q) -> ~ R) <-> ((Q ^ R) -> ~ P)); ! *4.15 (GB 281->277) ~ (((~ (P -> ~ Q) -> R) -> (~ (Q -> R) -> ~ P)) -> ~ ((~ (S -> ~ T) -> ~ U) -> (~ (U -> ~ S) -> ~ T))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D 1D3DD2DD2D13DD2D1311DD2D1D2DD2D1DD2D1D2DD2D1211DD2D13DD2D1D2DD2DD2D13 DD2D131111DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2 D1311DD2D1DD2D1DD22D11DD2D112DD2D1D2D1DD2D13DD2D1DD22D2DD2D13DD2D1311 3; ! 277 steps (P <-> P); ! *4.2 ~ ((P -> P) -> ~ (Q -> Q)); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD211DD211; ! 35 steps ((P <-> Q) <-> (Q <-> P)); ! *4.21 (GB 183->163) ~ ((~ (P -> ~ Q) -> ~ (Q -> ~ P)) -> ~ (~ (R -> ~ S) -> ~ (S -> ~ R))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D1DD22D2D D2D13DD2D131DD2D13DD2D131D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D1DD22D2D D2D13DD2D131DD2D13DD2D131; ! 163 steps (((P <-> Q) ^ (Q <-> R)) -> (P <-> R)); ! *4.22 (GB 329->273) (~ (~ ((P -> Q) -> ~ (R -> S)) -> ~ ~ ((Q -> T) -> ~ (U -> R))) -> ~ ((P -> T) -> ~ (U -> S))); ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2DD2D1DD2D121D3DD2D1D3D3DD2D 1D3DD2DD2D13DD2D13DD2D13DD2D13111DD2D131D3DD2D1DD2D1D3DD2DD2D13DD2D13 11DD2D1D2DD2D1311DD2D131DD2DD2D1DD2D121D3DD2D1DD2D1D3DD2DD2D13DD2D131 1DD2D1D2DD2D13111D3DD2D1D3D3DD2D1D3DD2DD2D13DD2D13DD2D13DD2D131111 ; ! 273 steps (P <-> (P ^ P)); ! *4.24 (GB 91->89) ~ ((P -> ~ (P -> ~ P)) -> ~ (~ (Q -> ~ R) -> R)); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1D231DD2D1D2DD2DD2D13DD2D13111D3DD2D1 D3DD2DD2D13DD2D13111; ! 89 steps (P <-> (P v P)); ! *4.25 ~ ((P -> (Q -> P)) -> ~ ((~ R -> R) -> R)); ! Result of proof DD3DD2DD2DD2D13DD2D1311D11DD2DD2D13D2DD2D1311; ! 45 steps ((P ^ Q) <-> (Q ^ P)); ! *4.3 (GB 183->163) ~ ((~ (P -> ~ Q) -> ~ (Q -> ~ P)) -> ~ (~ (R -> ~ S) -> ~ (S -> ~ R))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D1DD22D2D D2D13DD2D131DD2D13DD2D131D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D1DD22D2D D2D13DD2D131DD2D13DD2D131; ! 163 steps ((P v Q) <-> (Q v P)); ! *4.31 ~ (((~ P -> Q) -> (~ Q -> P)) -> ~ ((~ R -> S) -> (~ S -> R))) ; ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D13D2D1D3DD2 DD2D13DD2D1311; ! 83 steps (((P ^ Q) ^ R) <-> (P ^ (Q ^ R))); ! *4.32 (GB 359->355) ~ ((~ (~ (P -> ~ Q) -> R) -> ~ (P -> ~ ~ (Q -> R))) -> ~ (~ (S -> ~ ~ (T -> U)) -> ~ (~ (S -> ~ T) -> U))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD2D1DD2 D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112D2D1DD2D1DD2D13DD2D 1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311D2D1DD2DD2D13DD2D1311DD2D D2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD2D1D2D1D3DD2DD2D13DD2D1 311DD2D1D2DD2D1DD2D1D2DD2D1211DD2D13DD2D1D2DD2DD2D13DD2D131111DD2DD2D 13DD2D1311; ! 355 steps (((P v Q) v R) <-> (P v (Q v R))); ! *4.33 ~ (((~ (P -> Q) -> R) -> (P -> (~ Q -> R))) -> ~ ((S -> (~ T -> U)) -> (~ (S -> T) -> U))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1D2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1D D2D1DD22D11DD2D112DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D13D2D1D3DD2DD 2D13DD2D1311DD2D1DD2D1DD22D11DD2D112D2D1DD2D13D2D1D3DD2DD2D13DD2D1311 ; ! 207 steps ((P <-> Q) -> ((P ^ R) <-> (Q ^ R))); ! *4.36 (GB 311->291) (~ ((P -> Q) -> ~ (R -> S)) -> ~ ((~ (P -> T) -> ~ (Q -> T)) -> ~ (~ (R -> U) -> ~ (S -> U)))); ! Result of proof D3DDD22D2DD2D13DD2D131D1DD2D1D3DD2DD2D13DD2D1311DDDD2DD2D12DD2D11DD2D 12DD2D11DD2D121D1DD2D1D2DD2D1211D3DDD22D2DD2D13DD2D131D1DD2D1D3DD2DD2 D13DD2D1311DD2D1DD2D1DD2D1DD2133D2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D12D D2D13DD2D1311DD2D1DD2D1DD2D1DD2133D2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D1 2DD2D13DD2D1311; ! 291 steps ((P <-> Q) -> ((P v R) <-> (Q v R))); ! *4.37 (GB 311->307) (~ ((P -> Q) -> ~ (R -> S)) -> ~ (((~ P -> T) -> (~ Q -> T)) -> ~ ((~ R -> U) -> (~ S -> U)))); ! Result of proof D3DDD22D2DD2D13DD2D131D1DD2D1D3DD2DD2D13DD2D1311DDDD2DD2D12DD2D11DD2D 12DD2D11DD2D121D1DD2D1D2DD2D1211D3DDD22D2DD2D13DD2D131D1DD2D1D3DD2DD2 D13DD2D1311DD2D1DD2D1D2DD2D1211DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2 D1D3DD2DD2D13DD2D1311DD2D1DD2D1D2DD2D1211DD2D1DD2D13DD2D1DD22D2DD2D13 DD2D1311D2D1D3DD2DD2D13DD2D1311; ! 307 steps (((P <-> R) ^ (Q <-> S)) -> ((P ^ Q) <-> (R ^ S))); ! *4.38 (~ (~ ((P -> Q) -> ~ (R -> S)) -> ~ ~ ((T -> U) -> ~ (V -> W))) -> ~ ((~ (P -> ~ T) -> ~ (Q -> ~ U)) -> ~ (~ (R -> ~ V) -> ~ (S -> ~ W)))); ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2DD2D1DD2D12D2D1DD2D13DD2D1D 2DD2DD2D13DD2D13111DD2D1DD2D1DD2DD2D121D1D3DD2D1D3DD2DD2D13DD2D1311DD 2D131D3DD2D1D3DD2DD2D13DD2D1311DD2D131D3DD2D1D3DD2DD2D13DD2D1311DD2D1 31DD2D1DD2D1DD2DD2D121D1D3DD2D1D3DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13 DD2D1311DD2D131D3DD2D1D3DD2DD2D13DD2D13111DD2DD2D1DD2D12D2D1DD2D13DD2 D1D2DD2DD2D13DD2D13111DD2D1DD2D1DD2DD2D121D1D3DD2D1D3DD2DD2D13DD2D131 1DD2D131D3DD2D1D3DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2D1311DD2D131D D2D1DD2D1DD2DD2D121D1D3DD2D1D3DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2 D13111D3DD2D1D3DD2DD2D13DD2D13111; ! 585 steps (((P <-> R) ^ (Q <-> S)) -> ((P v Q) <-> (R v S))) ; ! *4.39 (GB 913->901) (~ (~ ((P -> Q) -> ~ (R -> S)) -> ~ ~ ((T -> U) -> ~ (V -> W))) -> ~ (((~ P -> T) -> (~ Q -> U)) -> ~ ((~ R -> V) -> (~ S -> W)))) ; ! Result of proof DD2D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1DDDD2DD2D12DD2D11DD2D12DD2D11DD2D 121D1DD2D1D2DD2D1211D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD2D1D2DD2D1DD2D13 D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112DD2D1D2D1DD2D1DD2D13DD 2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2DD2D121D1DD2D1D D2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1D2D1DD 2D1D2DD2D1211211DD2DD2D13DD2D1311DD2D1D2DD2D1DD2D13D2D1D3DD2DD2D13DD2 D1311DD2D1DD2D1DD22D11DD2D112DD2D1D2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2 D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2DD2D121D1DD2D1DD2D13DD2D1DD22D2DD 2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1D2D1DD2D1D2DD2D1211211DD 2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD2D1DD2D1D2D1D3DD 2DD2D13DD2D13113D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD2D1DD2D1DD22D11DD2 D112DD2D1DD2D1D2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13 DD2D1311D2D1DD2D1DD22D11DD2D112D2D13DD2D1DD22D11DD2D1123DD2DD2D13DD2D 1311; ! 901 steps ((P ^ (Q v R)) <-> ((P ^ Q) v (P ^ R))); ! *4.4 ~ ((~ (P -> ~ (Q -> R)) -> (~ ~ (P -> Q) -> ~ (P -> ~ R))) -> ~ ((~ ~ (S -> T) -> ~ (S -> ~ U)) -> ~ (S -> ~ (T -> U)))) ; ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DDD22D11D1D2DD2D1DD2D1DD2D13DD2D1DD22D2DD2D1 3DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2D12D2D1DD2D13DD2D1D2D1D3DD2DD 2D13DD2D1311DD2D1D2DD2DD2D13DD2D13111DD2DD2D13DD2D1311D3DD2D1DD2DD2D1 DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1D3DD2DD2D13DD2D1311D2D1D3DD2D1D3D D2DD2D13DD2D1311DD2D131D2D1D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D131DD2 DD2D13DD2D1311; ! 359 steps ((P v (Q ^ R)) <-> ((P v Q) ^ (P v R))); ! *4.41 ~ (((P -> ~ (Q -> ~ R)) -> ~ ((P -> Q) -> ~ (P -> R))) -> ~ (~ ((S -> T) -> ~ (S -> U)) -> (S -> ~ (T -> ~ U)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111D2D1D 3DD2D1D3DD2DD2D13DD2D1311DD2D131D2D1D3DD2D1D3DD2DD2D13DD2D13111D3DD2D 1D3DD2DD2D13DD2D1311DDD22D11D1D2DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2 D1DD2D1DD22D2DD2D13DD2D1311DD2D12D2D1DD2D13DD2D1D2DD2DD2D13DD2D13111 ; ! 275 steps (P <-> ((P ^ Q) v (P ^ ~ Q))); ! *4.42 (GB 175->165) ~ ((P -> (~ ~ (P -> Q) -> ~ (P -> ~ Q))) -> ~ ((~ ~ (R -> S) -> ~ (R -> T)) -> R)); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D1DD22D2DD2D13DD2D1311DD2D1D2D2DD2D1 3DD2D1D2DD2DD2D13DD2D131111D3DD2DD2D1DD2D1D2DD2D13DD2D1D2DD2D13DD2DD2 D13DD2D131111DD2D131DD2D131; ! 165 steps (P <-> ((P v Q) ^ (P v ~ Q))); ! *4.43 ~ ((P -> ~ ((~ P -> Q) -> ~ (~ P -> R))) -> ~ (~ ((~ S -> T) -> ~ (~ S -> ~ T)) -> S)); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1 D2DD2D1311DD2D1D2DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D2DD2D13DD2 D1D2DD2DD2D13DD2D131111; ! 161 steps (P <-> (P v (P ^ Q))); ! *4.44 ~ ((P -> (~ P -> Q)) -> ~ ((~ R -> ~ (R -> S)) -> R)) ; ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1D2DD2D1311DD2DD2D13D2DD2D1D2DD2D131DD2D 11DD2D1313; ! 79 steps (P <-> (P ^ (P v Q))); ! *4.45 (GB 161->107) ~ ((P -> ~ (P -> ~ (~ P -> Q))) -> ~ (~ (R -> S) -> R)) ; ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1D2DD2 D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D131; ! 107 steps ((P ^ Q) <-> ~ (~ P v ~ Q)); ! *4.5 (GB 177->161) ~ ((~ (P -> Q) -> ~ (~ ~ P -> Q)) -> ~ (~ (~ ~ R -> S) -> ~ (R -> S))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DDD22D1D1D3DD2DD2D 13DD2D1311DD2D12DD2D13DD2D131D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD22D2DD2 D13DD2D131DD2D13DD2D131; ! 161 steps (~ (P ^ Q) <-> (~ P v ~ Q)); ! *4.51 (GB 127->117) ~ ((~ ~ (P -> Q) -> (~ ~ P -> Q)) -> ~ ((~ ~ R -> S) -> ~ ~ (R -> S))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD22D2DD2D13DD2D131DD2D13DD2D131DD2D1D3 DD2DD2D13DD2D1311DD2DD2D121D1D3DD2DD2D13DD2D1311; ! 117 steps ((P ^ ~ Q) <-> ~ (~ P v Q)); ! *4.52 (GB 231->219) ~ ((~ (P -> ~ ~ Q) -> ~ (~ ~ P -> Q)) -> ~ (~ (~ ~ R -> S) -> ~ (R -> ~ ~ S))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD2D13DD2D13D 2D1D3DD2DD2D13DD2D13DD2D13DD2D1311DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13 DD2D1311DD2D1DD2D1D2D1DD2DD2D13DD2D1311DD2D1DD22D2DD2D13DD2D1311DD2DD 2D13DD2D1311; ! 219 steps (~ (P ^ ~ Q) <-> (~ P v Q)); ! *4.53 (GB 181->169) ~ ((~ ~ (P -> ~ ~ Q) -> (~ ~ P -> Q)) -> ~ ((~ ~ R -> S) -> ~ ~ (R -> ~ ~ S))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D1D2D1DD2DD2D13DD2D1311DD2D1DD22D2DD 2D13DD2D1311DD2DD2D13DD2D1311DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D13D2D1 D3DD2DD2D13DD2D13DD2D13DD2D1311; ! 169 steps ((~ P ^ Q) <-> ~ (P v ~ Q)); ! *4.54 ~ ((P -> P) -> ~ (Q -> Q)); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD211DD211; ! 35 steps (~ (~ P ^ Q) <-> (P v ~ Q)); ! *4.55 ~ ((~ ~ P -> P) -> ~ (Q -> ~ ~ Q)); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2DD2D13DD2D1311D3DD2DD2D13DD2D1311 ; ! 61 steps ((~ P ^ ~ Q) <-> ~ (P v Q)); ! *4.56 ~ ((~ (P -> ~ ~ Q) -> ~ (P -> Q)) -> ~ (~ (R -> S) -> ~ (R -> ~ ~ S))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1D3DD2DD2D 13DD2D1311DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1DD2DD2D 13DD2D1311DD2DD2D13DD2D1311; ! 165 steps (~ (~ P ^ ~ Q) <-> (P v Q)); ! *4.57 ~ ((~ ~ (P -> ~ ~ Q) -> (P -> Q)) -> ~ ((R -> S) -> ~ ~ (R -> ~ ~ S))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1D2D1DD2DD2D13DD2D1311DD2DD2D13DD2D1311D D2D1D3DD2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311; ! 115 steps ((P -> Q) <-> (~ P v Q)); ! *4.6 ~ (((P -> Q) -> (~ ~ P -> Q)) -> ~ ((~ ~ R -> S) -> (R -> S))) ; ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD22D2DD2D13DD2D1311DD2DD2D121D1D3DD2DD 2D13DD2D1311; ! 81 steps (~ (P -> Q) <-> (P ^ ~ Q)); ! *4.61 ~ ((~ (P -> Q) -> ~ (P -> ~ ~ Q)) -> ~ (~ (R -> ~ ~ S) -> ~ (R -> S))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1DD2DD2D13 DD2D1311DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1D3DD2DD2D 13DD2D1311DD2DD2D13DD2D1311; ! 165 steps ((P -> ~ Q) <-> (~ P v ~ Q)); ! *4.62 ~ (((P -> Q) -> (~ ~ P -> Q)) -> ~ ((~ ~ R -> S) -> (R -> S))) ; ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD22D2DD2D13DD2D1311DD2DD2D121D1D3DD2DD 2D13DD2D1311; ! 81 steps (~ (P -> ~ Q) <-> (P ^ Q)); ! *4.63 ~ ((P -> P) -> ~ (Q -> Q)); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD211DD211; ! 35 steps ((~ P -> Q) <-> (P v Q)); ! *4.64 ~ ((P -> P) -> ~ (Q -> Q)); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD211DD211; ! 35 steps (~ (~ P -> Q) <-> (~ P ^ ~ Q)); ! *4.65 ~ ((~ (P -> Q) -> ~ (P -> ~ ~ Q)) -> ~ (~ (R -> ~ ~ S) -> ~ (R -> S))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1DD2DD2D13 DD2D1311DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1D3DD2DD2D 13DD2D1311DD2DD2D13DD2D1311; ! 165 steps ((~ P -> ~ Q) <-> (P v ~ Q)); ! *4.66 ~ ((P -> P) -> ~ (Q -> Q)); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD211DD211; ! 35 steps (~ (~ P -> ~ Q) <-> (~ P ^ Q)); ! *4.67 ~ ((P -> P) -> ~ (Q -> Q)); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD211DD211; ! 35 steps ((P -> Q) <-> (P -> (P ^ Q))); ! *4.7 ~ (((P -> Q) -> (P -> ~ (P -> ~ Q))) -> ~ ((R -> ~ (S -> ~ T)) -> (R -> T))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D2DD2D13DD2D1D2DD2DD2D13DD2D13111D2D1D3DD2D1 D3DD2DD2D13DD2D13111; ! 89 steps ((P -> Q) <-> (P <-> (P ^ Q))); ! *4.71 ~ (((P -> Q) -> ~ ((P -> ~ (P -> ~ Q)) -> ~ (~ (R -> S) -> R))) -> ~ (~ ((T -> ~ (U -> ~ V)) -> W) -> (T -> V))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2DD2D13DD2D1D2DD2DD2D13DD2D13111D1D3D D2D1D3DD2DD2D13DD2D1311DD2D131D2DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1D 2D1D3DD2D1D3DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2D1311DD2D131 ; ! 201 steps ((P -> Q) <-> (Q <-> (P v Q))); ! *4.72 ~ (((P -> Q) -> ~ ((R -> (S -> R)) -> ~ ((~ P -> Q) -> Q))) -> ~ (~ (T -> ~ ((~ U -> V) -> W)) -> (U -> W))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1D3DD2DD2DD2D13DD2D1311D11DD2D1D2D1DD2DD 2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D121DD 2D1DD2DD2D121D1DD2D1D2DD2D1311D3DD2D1D3DD2DD2D13DD2D13111 ; ! 195 steps (Q -> (P <-> (P ^ Q))); ! *4.73 (P -> ~ ((Q -> ~ (Q -> ~ P)) -> ~ (~ (R -> S) -> R))) ; ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1D2DD2D13DD2D1D2DD2DD2D13D D2D131111D1D3DD2D1D3DD2DD2D13DD2D1311DD2D131; ! 113 steps (~ P -> (Q <-> (P v Q))); ! *4.74 (P -> ~ ((Q -> (R -> Q)) -> ~ ((P -> S) -> S))); ! Result of proof DD2D1D3DD2DD2DD2D13DD2D1311D11DD2D1D2DD2111; ! 43 steps (((P -> Q) ^ (P -> R)) <-> (P -> (Q ^ R))); ! *4.76 ~ ((~ ((P -> Q) -> ~ (P -> R)) -> (P -> ~ (Q -> ~ R))) -> ~ ((S -> ~ (T -> ~ U)) -> ~ ((S -> T) -> ~ (S -> U)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DDD22D11D1D2DD2D1D D2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D2DD2D13DD2D1311DD2D12D2D1D D2D13DD2D1D2DD2DD2D13DD2D13111DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111 D2D1D3DD2D1D3DD2DD2D13DD2D1311DD2D131D2D1D3DD2D1D3DD2DD2D13DD2D13111 ; ! 275 steps (((Q -> P) ^ (R -> P)) <-> ((Q v R) -> P)); ! *4.77 ~ ((~ ((P -> Q) -> ~ (R -> Q)) -> ((~ P -> R) -> Q)) -> ~ (((~ S -> T) -> U) -> ~ ((S -> U) -> ~ (T -> U)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D13DD2DD2D1DD2D12D2D1DD2D13DD2D1D2D1D3DD2 DD2D13DD2D1311DD2D1D2DD2DD2D13DD2D13111DD2D1DD2D1DD2D13DD2D1DD22D2DD2 D13DD2D1311D2D1D3DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD 2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311D3DD2D1 D3DD2DD2D13DD2D13111DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2DD2D121 D1DD2D1D2DD2D1311DD2DD2D121D11; ! 375 steps (((P -> Q) v (P -> R)) <-> (P -> (Q v R))); ! *4.78 (GB 327->319) ~ (((~ (P -> Q) -> (P -> R)) -> (P -> (~ Q -> R))) -> ~ ((S -> (~ T -> U)) -> (~ (V -> T) -> (S -> U)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D1DD2DD2D121D11DD2D12DD2DD2D121D11DD 2D1DD2D1DD22D11DD2D112DD2D1DD2D1DD2D1D2D13DD2D1DD2D1DD22D11DD2D112DD2 D1D2D1D2DD2D1D2D2DD2D13DD2D1D2DD2DD2D13DD2D131111DD2D1DD22D11DD2D112D 2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD22D11DD2D112DD2D1DD2DD2D121D1D D2D1DD2D1DD2D13DD22D1DD2D13DD2D1311DD2D1312; ! 319 steps (((Q -> P) v (R -> P)) <-> ((Q ^ R) -> P)); ! *4.79 (GB 547->383) ~ (((~ (P -> Q) -> (R -> Q)) -> (~ (P -> ~ R) -> Q)) -> ~ ((~ (S -> ~ T) -> U) -> (~ (S -> V) -> (T -> U)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1 DD2DD2D121D11DD2D12DD2DD2D121D11DD2D1D2D1DD2D1DD2D13DD2D1DD22D2DD2D13 DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112DD2D1DD2D1D2D1 DD2D1D2D1DD2DD2D13D2DD2D13112D2D1DD2D13DD2D1D2DD2D131D2D11DD2D1DD22D1 1DD2D112DD2D1DD2D1DD2D1DD2DD2D121D1DD2D1DD2D13DD2DD2D121D1DD2D131DD2D 131D2D132DD2D13D2D1D3DD2DD2D13DD2D1311; ! 383 steps ((P -> ~ P) <-> ~ P); ! *4.8 ~ (((P -> ~ P) -> ~ P) -> ~ (Q -> (R -> Q))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2DD2D13D2DD2D1311DD2D1DD22D2DD2D13DD2 D13111; ! 75 steps ((~ P -> P) <-> P); ! *4.81 ~ (((~ P -> P) -> P) -> ~ (Q -> (R -> Q))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2DD2D13D2DD2D13111; ! 45 steps (((P -> Q) ^ (P -> ~ Q)) <-> ~ P); ! *4.82 (GB 249->179) ~ ((~ ((P -> Q) -> ~ (P -> ~ Q)) -> ~ P) -> ~ (~ R -> ~ ((R -> S) -> ~ (R -> T)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DDD22D11D1D2DD2D1D D2D1DD22D2DD2D13DD2D1311D2DD2D13DD2D1D2DD2DD2D13DD2D13111DD2DD2D1DD2D 13DD2D1D2DD2DD2D13DD2D13111DD2D131DD2D131; ! 179 steps (((P -> Q) ^ (~ P -> Q)) <-> Q); ! *4.83 ~ ((~ ((P -> Q) -> ~ (~ P -> Q)) -> Q) -> ~ (R -> ~ ((S -> R) -> ~ (T -> R)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DDD22D11D1D2DD2D1D D2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1D2D1DD 2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D12 1DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D1311111; ! 249 steps ((P <-> Q) -> ((P -> R) <-> (Q -> R))); ! *4.84 (GB 139->135) (~ ((P -> Q) -> ~ (R -> S)) -> ~ (((S -> T) -> (R -> T)) -> ~ ((Q -> U) -> (P -> U)))); ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1D2DD2D121DD2D13DD2D1DD22D 11DD2D11DD2D131DD2D1D2DD2D121DD2D13DD2D1DD22D1DD2D131DD2D11DD2D131 ; ! 135 steps ((P <-> Q) -> ((R -> P) <-> (R -> Q))); ! *4.85 (~ ((P -> Q) -> ~ (R -> S)) -> ~ (((T -> P) -> (T -> Q)) -> ~ ((U -> R) -> (U -> S)))); ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1DD2D121D3DD2D1D3DD2DD2D13 DD2D1311DD2D131DD2D1DD2D121D3DD2D1D3DD2DD2D13DD2D13111; ! 123 steps ((P <-> Q) -> ((P <-> R) <-> (Q <-> R))); ! *4.86 (GB 621->617) (~ ((P -> Q) -> ~ (R -> S)) -> ~ ((~ ((S -> T) -> ~ (U -> P)) -> ~ ((R -> T) -> ~ (U -> Q))) -> ~ (~ ((Q -> V) -> ~ (W -> R)) -> ~ ((P -> V) -> ~ (W -> S))))); ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2DD2D1DD2D1D2D1DD2D1DD2D13DD 2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2DD2D121D1DD2D1D D2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D12DD2 D11DD2D121DD2D11DD2D1D2DD2D1211DD2D1D2DD2D121DD2D13DD2D1DD22D11DD2D11 DD2D131DD2D1DD2D121D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD2DD2D1DD2D1D2D1 DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2DD 2D121D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2 D1D2DD2D12DD2D11DD2D121DD2D11DD2D1D2DD2D1211DD2D1D2DD2D121DD2D13DD2D1 DD22D1DD2D131DD2D11DD2D131DD2D1DD2D121D3DD2D1D3DD2DD2D13DD2D13111 ; ! 617 steps (((((P ^ Q) -> R) <-> (P -> (Q -> R))) ^ ((P -> (Q -> R)) <-> (Q -> (P -> R)))) ^ ((Q -> (P -> R)) <-> ((Q ^ P) -> R))) ; ! *4.87 (GB 531->523) ~ (~ (~ (((~ (P -> ~ Q) -> R) -> (P -> (Q -> R))) -> ~ ((S -> (T -> U)) -> (~ (S -> ~ T) -> U))) -> ~ ~ (((V -> (W -> X)) -> (W -> (V -> X))) -> ~ ((Y -> (Z -> A)) -> (Z -> (Y -> A))))) -> ~ ~ (((B -> (C -> D)) -> (~ (B -> ~ C) -> D)) -> ~ ((~ (E -> ~ F) -> G) -> (E -> (F -> G))))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD3DD2DD2DD2D13DD2D1311D1DD3DD2DD2DD2D13DD2D 1311D1DD2D1D2DD2D1DD2D1D2DD2D1211DD2D13DD2D1D2DD2DD2D13DD2D131111DD2D 1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112D2D1DD2D1DD2D13 DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD3DD2DD2DD2D13DD2D13 11D1DD2D1DD22D11DD2D112DD2D1DD22D11DD2D112DD3DD2DD2DD2D13DD2D1311D1DD 2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112D2D1DD2D1DD2D 13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1D2DD2D1DD2D1D2 DD2D1211DD2D13DD2D1D2DD2DD2D13DD2D131111; ! 523 steps ((P ^ Q) -> (P <-> Q)); ! *5.1 (GB 111->107) (~ (P -> ~ Q) -> ~ ((R -> Q) -> ~ (S -> P))); ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1DD2D13DD22D11DD2D11DD2D13 1DD2D1DD2D13DD2DD2D121D1DD2D131DD2D131; ! 107 steps ((P -> Q) v (~ P -> Q)); ! *5.11 (GB 35->31) (~ (P -> Q) -> (~ P -> R)); ! Result of proof DD2D1DD22D1DD2D131DD2D11DD2D131; ! 31 steps ((P -> Q) v (P -> ~ Q)); ! *5.12 (GB 45->43) (~ (P -> Q) -> (R -> ~ Q)); ! Result of proof DD2D1DD2D1DD2D13DD22D1DD2D13DD2D1311DD2D131; ! 43 steps ((P -> Q) v (Q -> P)); ! *5.13 (GB 29->25) (~ (P -> Q) -> (Q -> R)); ! Result of proof DD2D1DD22D11DD2D11DD2D131; ! 25 steps ((P -> Q) v (Q -> R)); ! *5.14 (GB 29->25) (~ (P -> Q) -> (Q -> R)); ! Result of proof DD2D1DD22D11DD2D11DD2D131; ! 25 steps ((P <-> Q) v (P <-> ~ Q)); ! *5.15 (~ ~ ((P -> Q) -> ~ (R -> S)) -> ~ ((S -> ~ Q) -> ~ (~ R -> P))) ; ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1D2DD2D1DD2D1DD2D13DD2D1DD 22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1D2DD2D1DD2D13DD2D1 D2DD2DD2D13DD2D131111111DD2D1D2DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D 1DD2D1D2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1311DD2D1311 ; ! 267 steps ~ ((P <-> Q) ^ (P <-> ~ Q)); ! *5.16 (GB 377->333) ~ ~ (~ ((P -> Q) -> ~ (Q -> P)) -> ~ ~ ((P -> ~ Q) -> ~ (~ Q -> P))) ; ! Result of proof DD3DD2DD2D13DD2D1311DD2D1D3DD2DD2D13DD2D1311DD2DD2D1DD2D12D2D1DD2D13D D2D1D2D1D3DD2DD2D13DD2D1311DD2D1D2DD2DD2D13DD2D13111DD2D1DD2D1D2D1DD2 D1DD2DD2D13D2DD2D1311DD2D1DD22D2DD2D13DD2D1311DD2D1D2DD2D1211D3DD2D1D 3DD2DD2D13DD2D13111DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2DD2D13D D2D1D2DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2D1311DD2D131 ; ! 333 steps (((P v Q) ^ ~ (P ^ Q)) <-> (P <-> ~ Q)); ! *5.17 ~ ((~ ((~ P -> Q) -> ~ ~ ~ R) -> ~ (R -> ~ (~ Q -> P))) -> ~ (~ (S -> ~ (~ T -> U)) -> ~ ((~ U -> T) -> ~ ~ ~ S))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD2D1D2D1D3DD 2DD2D13DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D13DD 2D1DD22D2DD2D13DD2D1311DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2 D1DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311DD2DD2D121D1DD2D13D2D1D3DD 2DD2D13DD2D1311D2D1DD2DD2D13DD2D1311DD2DD2D13DD2D1311; ! 329 steps ((P <-> Q) <-> ~ (P <-> ~ Q)); ! *5.18 (GB 577->503) ~ ((~ ((P -> Q) -> ~ (Q -> P)) -> ~ ~ ((P -> ~ Q) -> ~ (~ Q -> P))) -> ~ (~ ~ ((R -> ~ S) -> ~ (~ T -> U)) -> ~ ((U -> S) -> ~ (T -> R)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1D3DD2DD2D13DD2D1311DD2DD2D1DD2D12D2D1DD 2D13DD2D1D2D1D3DD2DD2D13DD2D1311DD2D1D2DD2DD2D13DD2D13111DD2D1DD2D1D2 D1DD2D1DD2DD2D13D2DD2D1311DD2D1DD22D2DD2D13DD2D1311DD2D1D2DD2D1211D3D D2D1D3DD2DD2D13DD2D13111DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2DD 2D13DD2D1D2DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD2D1DD 2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1DD22D11DD2D11DD2D13DD2DD2D1 21D11DD2D1DD2D13DD2DD2D121D1DD2D131D2D1D3DD2D1D3DD2DD2D13DD2D1311DD2D 131DD2DD2D13DD2D1311; ! 503 steps ~ (P <-> ~ P); ! *5.19 (GB 141->131) ~ ~ ((P -> ~ P) -> ~ (~ P -> P)); ! Result of proof DD3DD2DD2D13DD2D1311DD2D1DD2DD2D13DD2D1DD2D1DD22D2DD2D13DD2D1311DD2D1 D2D2DD2D13111DD2D1DD2DD2D13D2DD2D1311DD2D1DD22D2DD2D13DD2D1311 ; ! 131 steps ((~ P ^ ~ Q) -> (P <-> Q)); ! *5.21 (GB 123->115) (~ (~ P -> ~ ~ Q) -> ~ ((P -> R) -> ~ (Q -> S))); ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2DD2D12DD2D11DD2D131D1DD2D1D 2DD2D1311DD2D13DD2D13DD2D1DD22D11DD2D11DD2D131; ! 115 steps (~ (P <-> Q) <-> ((P ^ ~ Q) v (Q ^ ~ P))); ! *5.22 ~ ((~ ~ ((P -> Q) -> ~ (R -> S)) -> (~ ~ (P -> ~ ~ Q) -> ~ (R -> ~ ~ S))) -> ~ ((~ ~ (T -> ~ ~ U) -> ~ (V -> ~ ~ W)) -> ~ ~ ((T -> U) -> ~ (V -> W)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D1DD2D1DD22D2DD2D13DD2D1311DD2D1DD2D D2D121D1D2D1DD2DD2D13DD2D1311D2D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1D D2DD2D13DD2D1311DD2DD2D13DD2D1311DD2DD2D13DD2D1311DD2D1D3DD2DD2D13DD2 D1311DD2D1DD2D1D2D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1D3DD2DD2D13DD2D 1311DD2DD2D13DD2D1311DD2DD2D121D1D2D1D3DD2DD2D13DD2D1311DD2DD2D121D1D 3DD2DD2D13DD2D1311; ! 363 steps ((P <-> Q) <-> ((P ^ Q) v (~ P ^ ~ Q))); ! *5.23 (GB 557->513) ~ ((~ ((P -> Q) -> ~ (Q -> P)) -> (~ ~ (P -> ~ Q) -> ~ (~ P -> ~ ~ Q))) -> ~ ((~ ~ (R -> ~ S) -> ~ (~ T -> ~ ~ U)) -> ~ ((T -> S) -> ~ (U -> R)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D1DD22D2DD2D13DD2D1311DD2DD2D1DD2D12 D2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13D D2D1311D2DD2D13DD2D1D2DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2D1311DD2 D131DD2D1DD2D1D2D1DD2D1DD2DD2D13D2DD2D1311DD2D1DD22D2DD2D13DD2D1311DD 2D1D2DD2D1211D3DD2D1D3DD2DD2D13DD2D13111DD2D1DD2DD2D1DD2D13DD2D1D2DD2 DD2D13DD2D13111DD2D1DD2D13DD2DD2D121D11D2D1D3DD2D1D3DD2DD2D13DD2D1311 DD2D131DD2D1DD2D13DD2DD2D121D1DD2D131D2D1D3DD2D1D3DD2DD2D13DD2D13111D D2DD2D121D1D3DD2DD2D13DD2D1311; ! 513 steps (~ ((P ^ Q) v (~ P ^ ~ Q)) <-> ((P ^ ~ Q) v (Q ^ ~ P))) ; ! *5.24 (GB 669->585) ~ ((~ (~ ~ (P -> ~ Q) -> ~ (~ P -> ~ ~ Q)) -> (~ ~ (P -> ~ ~ Q) -> ~ (Q -> ~ ~ P))) -> ~ ((~ ~ (R -> ~ S) -> ~ (T -> ~ ~ U)) -> ~ (~ ~ (U -> S) -> ~ (~ R -> ~ ~ T)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2D1DD22D2DD2D13DD2D1311DD2D1DD2DD2D12 1D1D2D1DD2DD2D13DD2D1311DD2DD2D1DD2D12D2D1DD2D13DD2D1D2DD2DD2D13DD2D1 3111DD2D1DD2D1DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1D2DD2D12113D3DD2D1D3DD 2DD2D13DD2D13111DD2D1DD2D1DD2D1D2D1DD2D1DD2DD2D13D2DD2D1311DD2D1DD22D 2DD2D13DD2D1311DD2D12D2D1DD2D131DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD 2D1311DD2D131DD2D1DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1DD2D1DD 2D1D3DD2DD2D13DD2D13113D2D1D3DD2D1D3DD2DD2D13DD2D13111DD2DD2D121D11DD 2D1D2D1D3DD2D1D3DD2DD2D13DD2D1311DD2D13DD2D13DD2D131DD2DD2D121D1DD2D1 31DD2DD2D121D1D3DD2DD2D13DD2D1311; ! 585 steps ((P v Q) <-> ((P -> Q) -> Q)); ! *5.25 ~ (((~ P -> Q) -> ((P -> Q) -> Q)) -> ~ (((R -> S) -> T) -> (~ R -> T))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1D2DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2D D2D121D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1211DD2DD2D121D1DD2D131 ; ! 133 steps (((P ^ Q) -> R) <-> ((P ^ Q) -> (P ^ R))); ! *5.3 ~ (((~ (P -> Q) -> R) -> (~ (P -> Q) -> ~ (P -> ~ R))) -> ~ ((S -> ~ (T -> ~ U)) -> (S -> U))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111D3DD2D 1D3DD2DD2D13DD2D1311DD2D131D2D1D3DD2D1D3DD2DD2D13DD2D13111 ; ! 127 steps ((R ^ (P -> Q)) -> (P -> (Q ^ R))); ! *5.31 (~ (P -> ~ (Q -> R)) -> (Q -> ~ (R -> ~ P))); ! Result of proof DD2D1D2DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D13 11DD2D1DD2DD2D121D1DD2D13DD2D1DD22D2DD2D13DD2D1311DD2D12DD2D11DD2D13D D2D1D2D1D3DD2DD2D13DD2D1311DD2D1D2DD2DD2D13DD2D131111; ! 191 steps ((P -> (Q <-> R)) <-> ((P ^ Q) <-> (P ^ R))); ! *5.32 (GB 633->625) ~ (((P -> ~ ((Q -> R) -> ~ (S -> T))) -> ~ ((~ (P -> ~ Q) -> ~ (P -> ~ R)) -> ~ (~ (P -> ~ S) -> ~ (P -> ~ T)))) -> ~ (~ ((~ (U -> ~ V) -> ~ (W -> ~ X)) -> ~ (~ (U -> ~ Y) -> ~ (Z -> ~ A))) -> (U -> ~ ((V -> X) -> ~ (Y -> A))))); ! Result of proof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steps ((P ^ (Q -> R)) <-> (P ^ ((P ^ Q) -> R))); ! *5.33 ~ ((~ (P -> ~ (Q -> R)) -> ~ (P -> ~ (~ (S -> ~ Q) -> R))) -> ~ (~ (T -> ~ (~ (T -> ~ U) -> V)) -> ~ (T -> ~ (U -> V)))) ; ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111D3DD2 D1D3DD2DD2D13DD2D1311DD2D131DD2D1DD2DD2D121D1D3DD2D1D3DD2DD2D13DD2D13 111D3DD2D1D3DD2DD2D13DD2D13111DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111 D3DD2D1D3DD2DD2D13DD2D1311DD2D131D3DD2D1D3DD2DD2D13DD2D1311DDD22D11D1 D2DD2D1DD2D1DD2D13DD2D1DD22D2DD2D13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2 D1DD2D1D2DD2D1211DD2D13DD2D1D2DD2DD2D13DD2D13111; ! 393 steps (((P -> Q) ^ (P -> R)) -> (P -> (Q <-> R))); ! *5.35 (~ ((P -> Q) -> ~ (P -> R)) -> (P -> ~ ((S -> R) -> ~ (T -> Q)))) ; ! Result of proof D3DD2D1D3DD2DD2D13DD2D1311DDD22D11D1D2DD2D1DD2D1DD2D13DD2D1DD22D2DD2D 13DD2D1311D2D1D3DD2DD2D13DD2D1311DD2D12D2D1DD2D1DD2D1D2DD2D1DD2D13DD2 D1D2DD2DD2D13DD2D13111111; ! 163 steps ((P ^ (P <-> Q)) <-> (Q ^ (P <-> Q))); ! *5.36 (GB 417->393) ~ ((~ (P -> ~ ~ ((P -> Q) -> R)) -> ~ (Q -> ~ ~ ((P -> Q) -> R))) -> ~ (~ (S -> ~ ~ (T -> ~ (S -> U))) -> ~ (U -> ~ ~ (T -> ~ (S -> U))))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111D3DD2 D1D3DD2DD2D13DD2D1311DDDDD2D1D2D112D1D2D1D3DD2DD2D13DD2D1311DD2D1DD2D 1DD22D11DD2D112DD2D12DD2D11DD2D1311D3DD2D1D3DD2DD2D13DD2D13111DD2DD2D 1DD2D13DD2D1D2DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2D1311DD2D1D2D1DD 2D1D3DD2DD2D13DD2D13111DD2D1D2DD2D13DD2D1D2D1D3DD2DD2D13DD2D1311DD2D1 D2DD2DD2D13DD2D131111D3DD2D1D3DD2DD2D13DD2D13111; ! 393 steps ((P -> (P -> Q)) <-> (P -> Q)); ! *5.4 ~ (((P -> (P -> Q)) -> (P -> Q)) -> ~ (R -> (S -> R))) ; ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD22D211; ! 33 steps (((P -> Q) -> (P -> R)) <-> (P -> (Q -> R))); ! *5.41 (GB 67->63) ~ ((((P -> Q) -> (R -> S)) -> (R -> (Q -> S))) -> ~ ((T -> (U -> V)) -> ((T -> U) -> (T -> V)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1DD2DD2D121D11DD2D12DD2DD2D121D112 ; ! 63 steps ((P -> (Q -> R)) <-> (P -> (Q -> (P ^ R)))); ! *5.42 (GB 109->105) ~ (((P -> (Q -> R)) -> (P -> (Q -> ~ (P -> ~ R)))) -> ~ ((S -> (T -> ~ (U -> ~ V))) -> (S -> (T -> V)))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D2DD2D1DD2D121DD2D13DD2D1D2DD2DD2D13DD2D1311 1D2D1D2D1D3DD2D1D3DD2DD2D13DD2D13111; ! 105 steps ((P -> Q) -> ((P -> R) <-> (P -> (Q ^ R)))); ! *5.44 ((P -> Q) -> ~ (((P -> R) -> (P -> ~ (Q -> ~ R))) -> ~ ((S -> ~ (T -> ~ U)) -> (S -> U)))); ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D12D2D1DD2D13DD2D1D2DD2DD2D 13DD2D13111D1D2D1D3DD2D1D3DD2DD2D13DD2D13111; ! 113 steps (P -> ((P -> Q) <-> Q)); ! *5.5 (P -> ~ (((P -> Q) -> Q) -> ~ (R -> (S -> R)))); ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1D2DD2111D11; ! 55 steps (P -> (Q <-> (P <-> Q))); ! *5.501 (P -> ~ ((Q -> ~ ((R -> Q) -> ~ (S -> P))) -> ~ (~ ((P -> T) -> U) -> T))); ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1DD2D1D2DD2D1DD2D13DD2D1D2 DD2DD2D13DD2D13111111DD2D1D2D3DD2D1D3DD2DD2D13DD2D1311DD2D1311 ; ! 131 steps ((((P v Q) v R) -> S) <-> (((P -> S) ^ (Q -> S)) ^ (R -> S))) ; ! *5.53 ~ ((((~ (~ P -> Q) -> R) -> S) -> ~ (~ ((P -> S) -> ~ (Q -> S)) -> ~ (R -> S))) -> ~ (~ (~ ((T -> U) -> ~ (V -> U)) -> ~ (W -> U)) -> ((~ (~ T -> V) -> W) -> U))); ! Result of proof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steps (((P ^ Q) <-> P) v ((P ^ Q) <-> Q)); ! *5.54 (GB 253->239) (~ ~ ((P -> Q) -> ~ (R -> ~ (R -> ~ S))) -> ~ ((~ (T -> ~ U) -> U) -> ~ (S -> P))); ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111D1D3DD2D1D3DD2DD2D13DD2D13111D 3DD2D1D3DD2DD2D13DD2D1311DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1 DD22D11DD2D11DD2D131DD2D1DD2D1D2DD2D13DD2D1D2DD2DD2D13DD2D13111D2D1D3 DD2D1D3DD2DD2D13DD2D1311DD2D1311; ! 239 steps (((P v Q) <-> P) v ((P v Q) <-> Q)); ! *5.55 (~ ~ (((~ P -> Q) -> P) -> ~ (R -> (~ R -> S))) -> ~ ((T -> Q) -> ~ (U -> (V -> U)))); ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13DD2D1311DD2D D2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1D2DD2D13D2D1DD2D1D3DD2DD2D13D D2D131111D1DD2D1D2DD2D1311D11; ! 167 steps (((P ^ ~ Q) -> R) <-> (P -> (Q v R))); ! *5.6 (GB 207->203) ~ (((~ (P -> ~ Q) -> R) -> (P -> (Q -> R))) -> ~ ((S -> (~ T -> U)) -> (~ (S -> ~ ~ T) -> U))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2D1D2DD2D1DD2D1D2DD2D1211DD2D13DD2D1D2DD2D D2D13DD2D131111DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D11DD2 D112D2D1DD2D1DD2D1D2D1D3DD2DD2D13DD2D13113D2D1D3DD2DD2D13DD2D1311 ; ! 203 steps (((P v Q) ^ ~ Q) <-> (P ^ ~ Q)); ! *5.61 (GB 269->259) ~ ((~ ((~ P -> Q) -> ~ ~ Q) -> ~ (P -> ~ ~ Q)) -> ~ (~ (R -> S) -> ~ ((~ R -> T) -> S))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D3DD2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD2DD2D1 21D1D2D1D3DD2DD2D13DD2D1311DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D12 1D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D121DD2DD2D13DD2D1311D3DD2D1D3DD2D D2D13DD2D1311DD2DD2D12DD2D13DD2D131D1DD2D1D2DD2D1311; ! 259 steps (((P ^ Q) v ~ Q) <-> (P v ~ Q)); ! *5.62 ~ (((~ ~ (P -> Q) -> R) -> (~ P -> R)) -> ~ ((~ S -> T) -> (~ ~ (S -> T) -> T))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2DD2D121D1DD2D1D3DD2DD2D13DD2D1311DD2D131D D2D1DD2D1DD22D2DD2D13DD2D1311DD2D1D2DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1 DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1211; ! 187 steps ((P v Q) <-> (P v (~ P ^ Q))); ! *5.63 ~ (((P -> Q) -> (P -> ~ (P -> ~ Q))) -> ~ ((R -> ~ (S -> ~ T)) -> (R -> T))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1D2DD2D13DD2D1D2DD2DD2D13DD2D13111D2D1D3DD2D1 D3DD2DD2D13DD2D13111; ! 89 steps (((P v R) <-> (Q v R)) <-> (R v (P <-> Q))); ! *5.7 (GB 681->673) ~ ((~ (((~ P -> Q) -> (~ R -> S)) -> ~ ((~ T -> U) -> (~ V -> S))) -> (~ S -> ~ ((P -> R) -> ~ (T -> V)))) -> ~ ((~ W -> ~ ((X -> Y) -> ~ (Z -> A))) -> ~ (((~ X -> W) -> (~ Y -> W)) -> ~ ((~ Z -> W) -> (~ A -> W))))); ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1DD2D12D2D1DD2D13DD2D1D2DD2DD2D13DD2D 13111DD2D1DD2D1DD2D1DD2D1DD2DD2D121D11DD2D12DD2DD2D121D11D2D1DD2D13D2 D1D3DD2DD2D13DD2D1311DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2D1311D3DD2D1 D3DD2DD2D13DD2D1311DD2D131DD2D1DD2D1DD2D1DD2D1DD2DD2D121D11DD2D12DD2D D2D121D11D2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2DD2D121D1DD2D13D2D1D3DD2 DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D13111DD2DD2D1DD2D13DD2D1D2DD2DD2D 13DD2D13111DD2D1DD2D1DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2D1311D2 D1DD2D13D2D1D3DD2DD2D13DD2D13112D2D1D3DD2D1D3DD2DD2D13DD2D1311DD2D131 DD2D1DD2D1DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2D1311D2D1DD2D13D2D 1D3DD2DD2D13DD2D13112D2D1D3DD2D1D3DD2DD2D13DD2D13111; ! 673 steps ((Q -> ~ R) -> (((P v Q) ^ R) <-> (P ^ R))); ! *5.71 (GB 259->249) ((P -> ~ Q) -> ~ ((~ ((~ R -> P) -> ~ Q) -> ~ (R -> ~ Q)) -> ~ (~ (S -> T) -> ~ ((~ S -> U) -> T)))); ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2DD2D1DD2D12D2D1DD2D13DD2D1D 2DD2DD2D13DD2D13111DD2D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D1DD2D1DD22D1 1DD2D112DD2D121D1D3DD2D1D3DD2DD2D13DD2D13111D1D3DD2D1D3DD2DD2D13DD2D1 311DD2DD2D12DD2D13DD2D131D1DD2D1D2DD2D1311; ! 249 steps ((P -> (Q <-> R)) <-> ((P -> Q) <-> (P -> R))) ; ! *5.74 (GB 345->337) ~ (((P -> ~ ((Q -> R) -> ~ (S -> T))) -> ~ (((P -> Q) -> (P -> R)) -> ~ ((P -> S) -> (P -> T)))) -> ~ (~ (((U -> V) -> (W -> X)) -> ~ ((Y -> Z) -> (W -> A))) -> (W -> ~ ((V -> X) -> ~ (Z -> A))))) ; ! Result of proof DD3DD2DD2DD2D13DD2D1311D1DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1 2D2D1D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD2D12D2D1D3DD2D1D3DD2DD2D13DD2 D13111DD2DD2D1DD2D12D2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1DD2D1DD2D D2D121D11DD2D12DD2DD2D121D11D3DD2D1D3DD2DD2D13DD2D1311DD2D131DD2D1DD2 D1DD2DD2D121D11DD2D12DD2DD2D121D11D3DD2D1D3DD2DD2D13DD2D13111 ; ! 337 steps (((R -> ~ Q) ^ (P <-> (Q v R))) -> ((P ^ ~ Q) <-> R)); ! *5.75 (~ ((P -> Q) -> ~ ~ ((R -> (~ S -> T)) -> ~ ((U -> P) -> V))) -> ~ ((~ (R -> ~ ~ S) -> T) -> ~ (P -> ~ (V -> ~ Q)))) ; ! Result of proof DD2DD2D1DD2D13DD2D1D2DD2DD2D13DD2D13111DD2D1DD2D1DD2D1DD2D13D2D1D3DD2 DD2D13DD2D1311DD2D1DD2D1DD22D11DD2D112D2D1DD2D1DD2D1D2D1D3DD2DD2D13DD 2D13113D2D1D3DD2DD2D13DD2D1311D3DD2D1D3DD2DD2D13DD2D1311DD2D131D3DD2D 1D3DD2DD2D13DD2D13111DD2DD2D1DD2D12D2D1DD2D13DD2D1D2DD2DD2D13DD2D1311 1DD2D1DD2D1DD2DD2D121D11D3DD2D1D3DD2DD2D13DD2D13111D3DD2D1D3DD2DD2D13 DD2D13111D3DD2D1D3DD2DD2D13DD2D1311DD2D131; ! 387 steps (((((P -> Q) -> (~ R -> ~ S)) -> R) -> T) -> ((T -> P) -> (S -> P))) ; ! Meredith's single axiom (contributed by Scott Fenton 12-Jun-2010) (((((P -> Q) -> (~ R -> ~ S)) -> R) -> T) -> ((T -> P) -> (S -> P))) ; ! Result of proof DDDD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13D D2D1311DD2D121DD2D11DD2D111DD2DD2D1DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1D D2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D121DD2DD2D12DD2D11DD2D1DD 2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD2D13DD2D1 311DD2D121DD2D1D2D11DD2D1D2D1DD2D131DD2D1D2D1DD23D13DD2DD2D12DD2D1D2D 12DD2D1D2D11DD2D1D2DD2D1D22DD2D111DD2D11DD2D12DD2D11DD2D131D1D1DD2DD2 D13DD2D1311D1DD2D1D2D1DD2D131DD2D1D2D1DD23D13DD2DD2D12DD2D1D2D12DD2D1 D2D11DD2DD2D12DD2D11DD2D1D22DD2D11DD2D111D1DD2D12DD2D11DD2D131D1D1DD2 DD2D13DD2D1311DD2DD2D12DD2D1D2D1DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2D D2D121D1DD2D13D2D1D3DD2DD2D13DD2D1311DD2D121DD2DD2D12DD2D11DD2D12DD2D 11DD2D1DD2D1D2D1DD2DD2D13D2DD2D1311DD2D1DD2DD2D121D1DD2D13D2D1D3DD2DD 2D13DD2D1311DD2D121DD2D1D2D11DD2D1D2D1DD2D131DD2D1D2D1DD23D13DD2DD2D1 2DD2D1D2D12DD2D1D2D11DD2D1D2DD2D1D22DD2D111DD2D11DD2D12DD2D11DD2D131D 1D1DD2DD2D13DD2D1311DD2D1D2D1D2D1D2D1DD2D131DD2D1D2D1D2D1D2D1DD23D13D D2DD2D12DD2D1D2D12DD2D1D2D1D2D12DD2D1D2D1D2D1D2D12DD2D1D2D1D2D1D2D11D D2DD2D12DD2D1D2D12DD2D1D2D1D2D12DD2D1D2D1D2D11DD2D1D2D1D2D1D22DD2D1D2 D1D2D11DD2D1D2D1D2D11DD2D1D2D1D2D1DD2D131DD2D1D2D1D2D1DD23D13DD2DD2D1 2DD2D1D2D12DD2D1D2D1D2D12DD2D1D2D1D2D11DD2DD2D12DD2D11DD2D12DD2D11DD2 D1D22DD2D11DD2D11DD2D131D1DD2D1D2D12DD2D1D2D11DD2DD2D12DD2D11DD2D1D22 DD2D111D1DD2D12DD2D11DD2D13DD2D11D3DD2DD2D13DD2D1311D1D1D1DD2DD2D13DD 2D1311D1D1D1DD2D12DD2D11DD2D131D1D1D1D1DD2DD2D13DD2D1311D1D1DD2D11DD2 D11DD2D131; ! 1459 steps