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Type | Label | Description | ||||||||||||||||||||||||||||||
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Theorem | orbi1r 33401 | orbi1 705 with order of disjuncts reversed. Derived from orbi1rVD 33770. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | bitr3 33402 | Closed nested implication form of bitr3i 251. Derived automatically from bitr3VD 33771. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | 3orbi123 33403 | pm4.39 871 with a 3-conjunct antecedent. This proof is 3orbi123VD 33772 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | syl5imp 33404 | Closed form of syl5 32. Derived automatically from syl5impVD 33785. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | impexpd 33405 |
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. After the
User's Proof was completed, it was minimized. The completed User's Proof
before minimization is not shown. (Contributed by Alan Sare,
18-Mar-2012.) (Proof modification is discouraged.)
(New usage is discouraged.)
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Theorem | com3rgbi 33406 |
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual Deduction Proof (not shown) was minimized. The
minimized proof is shown.
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
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Theorem | impexpdcom 33407 |
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual Deduction Proof (not shown) was minimized. The
minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
| ||||||||||||||||||||||||||||||
Theorem | ee1111 33408 |
Non-virtual deduction form of e1111 33583. (Contributed by Alan Sare,
18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof
completed automatically by the tools program completeusersproof.cmd,
which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof
Assistant. The completed Virtual Deduction Proof (not shown) was
minimized. The minimized proof is shown.
| ||||||||||||||||||||||||||||||
Theorem | pm2.43bgbi 33409 |
Logical equivalence of a 2-left-nested implication and a 1-left-nested
implicated
when two antecedents of the former implication are identical.
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual
Deduction Proof (not shown) was minimized. The minimized proof is
shown.
| ||||||||||||||||||||||||||||||
Theorem | pm2.43cbi 33410 |
Logical equivalence of a 3-left-nested implication and a 2-left-nested
implicated when two antecedents of the former implication are identical.
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is
a Virtual Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof
(not shown) was minimized. The minimized proof is shown.
| ||||||||||||||||||||||||||||||
Theorem | ee233 33411 |
Non-virtual deduction form of e233 33684. (Contributed by Alan Sare,
18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual
Deduction Proof (not shown) was minimized. The minimized proof is
shown.
| ||||||||||||||||||||||||||||||
Theorem | imbi13 33412 | Join three logical equivalences to form equivalence of implications. imbi13 33412 is imbi13VD 33796 without virtual deductions and was automatically derived from imbi13VD 33796 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | ee33 33413 |
Non-virtual deduction form of e33 33653. (Contributed by Alan Sare,
18-Mar-2012.) (Proof modification is discouraged.)
(New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof
completed automatically by the tools program completeusersproof.cmd,
which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof
Assistant. The completed Virtual Deduction Proof (not shown) was
minimized. The minimized proof is shown.
| ||||||||||||||||||||||||||||||
Theorem | con5 33414 | Biconditional contraposition variation. This proof is con5VD 33822 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | con5i 33415 | Inference form of con5 33414. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | exlimexi 33416 | Inference similar to Theorem 19.23 of [Margaris] p. 90. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | sb5ALT 33417* | Equivalence for substitution. Alternate proof of sb5 2175. This proof is sb5ALTVD 33835 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | eexinst01 33418 | exinst01 33533 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | eexinst11 33419 | exinst11 33534 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | vk15.4j 33420 | Excercise 4j of Unit 15 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. This proof is the minimized Hilbert-style axiomatic version of the Fitch-style Natural Deduction proof found on page 442 of Klenk and was automatically derived from that proof. vk15.4j 33420 is vk15.4jVD 33836 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | notnot2ALT 33421 | Converse of double negation. Alternate proof of notnot2 112. This proof is notnot2ALTVD 33837 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | con3ALT 33422 | Contraposition. Alternate proof of con3 134. This proof is con3ALTVD 33838 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | ssralv2 33423* | Quantification restricted to a subclass for two quantifiers. ssralv 3560 for two quantifiers. The proof of ssralv2 33423 was automatically generated by minimizing the automatically translated proof of ssralv2VD 33788. The automatic translation is by the tools program translate_{without}_overwriting.cmd (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | sbc3or 33424 | sbcor 3372 with a 3-disjuncts. This proof is sbc3orgVD 33773 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | sbc3orgOLD 33425 | sbcorgOLD 3373 with a 3-disjuncts. This proof is sbc3orgVD 33773 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | alrim3con13v 33426* | Closed form of alrimi 1878 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 33774 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | rspsbc2 33427* | rspsbc 3413 with two quantifying variables. This proof is rspsbc2VD 33777 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | sbcoreleleq 33428* | Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 33781. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | tratrb 33429* | If a class is transitive and any two distinct elements of the class are E-comparable, then every element of that class is transitive. Derived automatically from tratrbVD 33783. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | ordelordALT 33430 | An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4909 using the Axiom of Regularity indirectly through dford2 8054. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that because this is inferred by the Axiom of Regularity. ordelordALT 33430 is ordelordALTVD 33789 without virtual deductions and was automatically derived from ordelordALTVD 33789 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | sbcim2g 33431 | Distribution of class substitution over a left-nested implication. Similar to sbcimg 3369. sbcim2g 33431 is sbcim2gVD 33797 without virtual deductions and was automatically derived from sbcim2gVD 33797 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | sbcbi 33432 | Implication form of sbcbiiOLD 3388. sbcbi 33432 is sbcbiVD 33798 without virtual deductions and was automatically derived from sbcbiVD 33798 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | trsbc 33433* | Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 33433 is trsbcVD 33799 without virtual deductions and was automatically derived from trsbcVD 33799 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | truniALT 33434* | The union of a class of transitive sets is transitive. Alternate proof of truni 4564. truniALT 33434 is truniALTVD 33800 without virtual deductions and was automatically derived from truniALTVD 33800 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | sbcssOLD 33435 | Distribute proper substitution through a subclass relation. This theorem was automatically derived from sbcssgVD 33805. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | onfrALTlem5 33436* | Lemma for onfrALT 33443. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | onfrALTlem4 33437* | Lemma for onfrALT 33443. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | onfrALTlem3 33438* | Lemma for onfrALT 33443. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | ggen31 33439* | gen31 33529 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | onfrALTlem2 33440* | Lemma for onfrALT 33443. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | cbvexsv 33441* | A theorem pertaining to the substitution for an existentially quantified variable when the substituted variable does not occur in the quantified wff. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | onfrALTlem1 33442* | Lemma for onfrALT 33443. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | onfrALT 33443 | The epsilon relation is foundational on the class of ordinal numbers. onfrALT 33443 is an alternate proof of onfr 4926. onfrALTVD 33813 is the Virtual Deduction proof from which onfrALT 33443 is derived. The Virtual Deduction proof mirrors the working proof of onfr 4926 which is the main part of the proof of Theorem 7.12 of the first edition of TakeutiZaring. The proof of the corresponding Proposition 7.12 of [TakeutiZaring] p. 38 (second edition) does not contain the working proof equivalent of onfrALTVD 33813. This theorem does not rely on the Axiom of Regularity. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | csbeq2gOLD 33444 | Formula-building implication rule for class substitution. Closed form of csbeq2i 3844. csbeq2gOLD 33444 is derived from the virtual deduction proof csbeq2gVD 33814. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete version of csbeq2 3434 as of 11-Oct-2018. (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | 19.41rg 33445 | Closed form of right-to-left implication of 19.41 1972, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 33824. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | opelopab4 33446* | Ordered pair membership in a class abstraction of pairs. Compare to elopab 4764. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | 2pm13.193 33447 | pm13.193 31501 for two variables. pm13.193 31501 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 33825. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | hbntal 33448 | A closed form of hbn 1896. hbnt 1895 is another closed form of hbn 1896. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | hbimpg 33449 | A closed form of hbim 1923. Derived from hbimpgVD 33826. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | hbalg 33450 | Closed form of hbal 1845. Derived from hbalgVD 33827. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | hbexg 33451 | Closed form of nfex 1949. Derived from hbexgVD 33828. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 12-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | ax6e2eq 33452* | Alternate form of ax6e 2003 for non-distinct , and . ax6e2eq 33452 is derived from ax6e2eqVD 33829. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | ax6e2nd 33453* | If at least two sets exist (dtru 4647) , then the same is true expressed in an alternate form similar to the form of ax6e 2003. ax6e2nd 33453 is derived from ax6e2ndVD 33830. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | ax6e2ndeq 33454* | "At least two sets exist" expressed in the form of dtru 4647 is logically equivalent to the same expressed in a form similar to ax6e 2003 if dtru 4647 is false implies . ax6e2ndeq 33454 is derived from ax6e2ndeqVD 33831. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | 2sb5nd 33455* | Equivalence for double substitution 2sb5 2188 without distinct , requirement. 2sb5nd 33455 is derived from 2sb5ndVD 33832. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | 2uasbanh 33456* | Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 33456 is derived from 2uasbanhVD 33833. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | 2uasban 33457* | Distribute the unabbreviated form of proper substitution in and out of a conjunction. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | e2ebind 33458 | Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 33458 is derived from e2ebindVD 33834. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | elpwgded 33459 | elpwgdedVD 33839 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | trelded 33460 | Deduction form of trel 4557. In a transitive class, the membership relation is transitive. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | jaoded 33461 | Deduction form of jao 512. Disjunction of antecedents. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | 3imp31 33462 | The importation inference 3imp 1190 with commutation of the first and third conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | 3imp21 33463 | The importation inference 3imp 1190 with commutation of the first and second conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | biimpa21 33464 | biimpa 484 with commutation of the first and second conjuncts of the assertion. (Contributed by Alan Sare, 11-Sep-2016.) | ||||||||||||||||||||||||||||||
Theorem | sbtT 33465 | A substitution into a theorem remains true. sbt 2163 with the existence of no virtual hypotheses for the hypothesis expressed as the empty virtual hypothesis collection. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | ex3 33466 | Apply ex 434 to a hypothesis with a 3-right-nested conjunction antecedent, with the antecedent of the assertion being a triple conjunction rather than a 2-right-nested conjunction. (Contributed by Alan Sare, 22-Apr-2018.) | ||||||||||||||||||||||||||||||
Theorem | not12an2impnot1 33467 | If a double conjunction is false and the second conjunct is true, then the first conjunct is false. http://us.metamath.org/other/completeusersproof/not12an2impnot1vd.html is the Virtual Deduction proof verified by automatically transforming it into the Metamath proof of not12an2impnot1 33467 using completeusersproof, which is verified by the Metamath program. http://us.metamath.org/other/completeusersproof/not12an2impnot1ro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. (Contributed by Alan Sare, 13-Jun-2018.) | ||||||||||||||||||||||||||||||
Syntax | wvd1 33468 |
A Virtual Deduction proof in a Hilbert-style deductive system is the
analog of a sequent calculus proof. A theorem is proven in a Gentzen
system in order to prove more directly, which may be more intuitive
and easier for some people. The analog of this proof in Metamath's
Hilbert-style system is verified by the Metamath program.
Natural Deduction is a well-known proof method orignally proposed by Gentzen in 1935 and comprehensively summarized by Prawitz in his 1965 monograph "Natural deduction: a proof-theoretical study". Gentzen wished to construct "a formalism that comes as close as possible to natural reasoning". Natural deduction is a response to dissatisfaction with axiomatic proofs such as Hilbert-style axiomatic proofs, which the proofs of Metamath are. In 1926, in Poland, Lukasiewicz advocated a more natural treatment of logic. Jaskowski made the earliest attempts at defining a more natural deduction. Natural deduction in its modern form was independently proposed by Gentzen. Sequent calculus, the chief alternative to Natural Deduction, was created by Gentzen. The following is an excerpt from Stephen Cole Kleene's seminal 1952 book "Introduction to Metamathematics", which contains the first formulation of sequent calculus in the modern style. Kleene states on page 440: . . . the proof (of Gentzen's Hauptsatz) breaks down into a list of cases, each of which is simple to handle. . . . Gentzen's normal form for proofs in the predicate calculus requires a different classification of the deductive steps than is given by the postulates of the formal system of predicate calculus of Chapter IV (Section 19). The implication symbol (the Metamath symbol for implication has been substituted here for the symbol used by Kleene) has to be separated in its role of mediating inferences from its role as a component symbol of the formula being proved. In the former role it will be replaced by a new formal symbol (read "gives" or "entails"), to which properties will be assigned similar to those of the informal symbol in our former derived rules. Gentzen's classification of the deductive operations is made explicit by setting up a new formal system of the predicate calculus. The formal system of propositional and predicate calculus studied previously (Chapters IV ff.) we call now a "Hilbert-type system", and denote by H. Precisely, H denotes any one or a particular one of several systems, according to whether we are considering propositional calculus or predicate calculus, in the classical or the intuitionistic version (Section 23), and according to the sense in which we are using "term" and "formula" (Sections 117,25,31,37,72-76). The same respective choices will apply to the "Gentzen-type system G1" which we introduce now and the G2, G3 and G3a later. The transformation or deductive rules of G1 will apply to objects which are not formulas of the system H, but are built from them by an additional formation rule, so we use a new term "sequent" for these objects. (Gentzen says "Sequenz", which we translate as "sequent", because we have already used "sequence" for any succession of objects, where the German is "Folge".) A sequent is a formal expression of the form , . . . , , . . . , where , . . . , and , . . . , are seqences of a finite number of 0 or more formulas (substituting Metamath notation for Kleene's notation). The part , . . . , is the antecedent, and , . . . , the succedent of the sequent , . . . , , . . . , . When the antecedent and the succedent each have a finite number of 1 or more formulas, the sequent , . . . , , . . . has the same interpretation for G1 as the formula . . . . . . for H. The interpretation extends to the case of an antecedent of 0 formulas by regarding . . . for 0 formulas (the "empty conjunction") as true and . . . for 0 formulas (the "empty disjunction") as false. . . . As in Chapter V, we use Greek capitals . . . to stand for finite sequences of zero or more formulas, but now also as antecedent (succedent), or parts of antecedent (succedent), with separating formal commas included. . . . (End of Kleene excerpt) In chapter V entitled "Formal Deduction" Kleene states, on page 86: Section 20. Formal deduction. Formal proofs of even quite elementary theorems tend to be long. As a price for having analyzed logical deduction into simple steps, more of those steps have to be used. The purpose of formalizing a theory is to get an explicit definition of what constitutes proof in the theory. Having achieved this, there is no need always to appeal directly to the definition. The labor required to establish the formal provability of formulas can be greatly lessened by using metamathematical theorems concerning the existence of formal proofs. If the demonstrations of those theorems do have the finitary character which metamathematics is supposed to have, the demonstrations will indicate, at least implicitly, methods for obtaining the formal proofs. The use of the metamathematical theorems then amounts to abbreviation, often of very great extent, in the presentation of formal proofs. The simpler of such metamathematical theorems we shall call derived rules, since they express principles which can be said to be derived from the postulated rules by showing that the use of them as additional methods of inference does not increase the class of provable formulas. We shall seek by means of derived rules to bring the methods for establishing the facts of formal provability as close as possible to the informal methods of the theory which is being formalized. In setting up the formal system, proof was given the simplest possible structure, consisting of a single sequence of formulas. Some of our derived rules, called "direct rules", will serve to abbreviate for us whole segments of such a sequence; we can then, so to speak, use these segments as prefabricated units in building proofs. But also, in mathematical practice, proofs are common which have a more complicated structure, employing "subsidiary deduction", i.e. deduction under assumptions for the sake of the argument, which assumptions are subsequently discharged. For example, subsidiary deduction is used in a proof by reductio ad absurdum, and less obtrusively when we place the hypothesis of a theorem on a par with proved propositions to deduce the conclusion. Other derived rules, called "subsidiary deduction rules", will give us this kind of procedure. We now introduce, by a metamathematical definition, the notion of "formal deducibility under assumptions". Given a list , . . . of or more (occurrences of) formulas, a finite sequence of one or more (occurrences of) formulas is called a (formal) deduction from the assumption formulas , . . . , if each formula of the sequence is either one of the formulas , . . . , or an axiom, or an immediate consequence of preceding formulas of a sequence. A deduction is said to be deducible from the assumption formulas (in symbols, ,. . . . ,. ), and is called the conclusion (or endformula) of the deduction. (The symbol may be read "yields".) (End of Kleene excerpt) Gentzen's normal form is a certain direct fashion for proofs and deductions. His sequent calculus, formulated in the modern style by Kleene, is the classical system G1. In this system, the new formal symbol has properties similar to the informal symbol of Kleene's above language of formal deducibility under assumptions. Kleene states on page 440: . . . This leads us to inquire whether there may not be a theorem about the predicate calculus asserting that, if a formula is provable (or deducible from other formulas), it is provable (or deducible) in a certain direct fashion; in other words, a theorem giving a normal form for proofs and deductions, the proofs and deduction in normal form being in some sense direct. (End of Kleene excerpt) There is such a theorem, which was proven by Kleene. Formal proofs in H of even quite elementary theorems tend to be long. As a price for having analyzed logical deduction into simple steps, more of those steps have to be used. The proofs of Metamath are fully detailed formal proofs. We wish to have a means of writing rigorously verifiable mathematical proofs in a more direct fashion. Natural Deduction is a system for proving theorems and deductions in a more direct fashion. However, Natural Deduction is not compatible for use with Metamath, which uses a Hilbert-type system. Instead, Kleene's classical system G1 may be used for proving Metamath deductions and theorems in a more direct fashion. The system of Metamath is an H system, not a Gentzen system. Therefore, proofs in Kleene's classical system G1 ("G1") cannot be included in Metamath's system H, which we shall henceforth call "system H" or "H". However, we may translate proofs in G1 into proofs in H. By Kleene's THEOREM 47 (page 446)
By Kleene's COROLLARY of THEOREM 47 (page 448)
denotes the same connective denoted by . " , " , in the context of Virtual Deduction, denotes the same connective denoted by . This Virtual Deduction notation is specified by the following set.mm definitions:
replaces in the analog in H of a sequent in G1 having a nonempty antecedent. If occurs as the outermost connective denoted by or and occurs exactly once, we call the analog in H of a sequent in G1 a "virtual deduction" because the corresponding of the sequent is assigned properties similar to . While sequent calculus proofs (proofs in G1) may have as steps sequents with 0, 1, or more formulas in the succedent, we shall only prove in G1 using sequents with exactly 1 formula in the succedent. The User proves in G1 in order to obtain the benefits of more direct proving using sequent calculus, then translates the proof in G1 into a proof in H. The reference theorems and deductions to be used for proving in G1 are translations of theorems and deductions in set.mm. Each theorem in set.mm corresponds to the theorem in G1. Deductions in G1 corresponding to deductions in H are similarly determined. Theorems in H with one or more occurrences of either or may also be translated into theorems in G1 for by replacing the outermost occurrence of or of the theorem in H with . Deductions in H may be translated into deductions in G1 in a similar manner. The only theorems and deductions in H useful for proving in G1 for the purpose of obtaining proofs in H are those in which, for each hypothesis or assertion, there are 0 or 1 occurrences of and it is the outermost occurrence of or . Kleene's THEOREM 46 and its COROLLARY 2 are used for translating from H to G1. By Kleene's THEOREM 46 (page 445)
By Kleene's COROLLARY 2 of THEOREM 46 (page 446)
To prove in H, the User simply proves in G1 and translates each G1-proof step into a H-proof step. The translation is trivial and immediate. The proof in H is in Virtual Deduction notation. It is a working proof in the sense that, if it has no errors, each theorem and deduction of the proof is true, but may or may not, after being translated into conventional notation, unify with any theorem or deduction scheme in set.mm. Each theorem or deduction scheme in set.mm has a particular form. The working proof written by the User (the "User's Proof" or "Virtual Deduction Proof") may contain theorems and deductions which would unify with a variant of a theorem or deduction scheme in set.mm, but not with any particular form of that theorem or deduction scheme in set.mm. The computer program completeusersproof.c may be applied to a Virtual Deduction proof to automatically add steps to the proof ("technical steps") which, if possible, transforms the form of a theorem or deduction of the Virtual Deduction proof not unifiable with a theorem or deduction scheme in set.mm into a variant form which is. For theorems and deductions of the Virtual Deduction proof which are completable in this way, completeusersproof saves the User the extra work involved in satisfying the constraint that the theorem or deduction is in a form which unifies with a theorem or deduction scheme in set.mm. mmj2, which is invoked by completeusersproof, automatically finds one of the reference theorems or deductions in set.mm which unifies with each theorem and deduction in the proof satisfying this constraint and labels the theorem or the assertion step of the deduction. The analogs in H of the postulates of G1 are the set.mm postulates. The postulates in G1 corresponding to the Metamath postulates are not the classical system G1 postulates of Kleene (pages 442 and 443). set.mm has the predicate calculus postulates and other posulates. The Kleene classical system G1 postulates correspond to predicate calculus postulates which differ from the Metamath system G1 postulates corresponding to the predicate calculus postulates of Metamath's system H. Metamath's predicate calculus G1 postulates are presumably deducible from the Kleene classical G1 postulates and the Kleene classical G1 postulates are deducible from Metamath's G1 postulates. It should be recognized that, because of the different postulates, the classical G1 system corresponding to Metamath's system H is not identical to Kleene's classical system G1. Why not create a separate database (setg.mm) of proofs in G1, avoiding the need to translate from H to G1 and from G1 back to H? The translations are trivial. Sequents make the language more complex than is necessary. More direct proving using sequent calculus may be done as a means towards the end of constructing proofs in H. Then, the language may be kept as simple as possible. A system G1 database would be redundant because it would duplicate the information contained in the corresponding system H database. For earlier proofs, each "User's Proof" in the web page description of a Virtual Deduction proof in set.mm is the analog in H of the User's working proof in G1. The User's Proof is automatically completed by completeusersproof.cmd (superseded by completeusersproof.c in September of 2016). The completed proof is the Virtual Deduction proof, which is the analog in H of the corresponding fully detailed proof in G1. The completed Virtual Deduction proof of these earlier proofs may be automatically translated into a conventional Metamath proof. The input for completeusersproof.c is a Virtual Deduction proof. Unlike completeusersproof.cmd, the completed proof is in conventional notation. completeusersproof.c eliminates the virtual deduction notation of the Virtual Deduction proof after utilizing the information it provides. Applying mmj2's unify command is essential to completeusersproof. The mmj2 program is invoked within the completeusersproof.c function mmj2Unify(). The original mmj2 program was written by Mel L. O'Cat. Mario Carneiro has enhanced it. mmj2Unify() is called multiple times during the execution of completeusersproof. A Virtual Deduction proof is a Metamath-specific version of a Natural Deduction Proof. In order for mmj2 to complete a Virtual Deduction proof it is necessary that each theorem or deduction of the proof is in a form which unifies with a theorem or deduction scheme in set.mm. completeusersproof weakens this constraint. The User may write a Virtual Deduction proof and automatically transform it into a complete Metamath proof using the completeusersproof tool. The completed proof has been checked by the Metamath program. The task of writing a complete Metamath proof is reduced to writing what is essentially a Natural Deduction Proof. The completeusersproof program and all associated files necessary to use it may be downloaded from the Metamath web site. All syntax definitions, theorems, and deductions necessary to create Virtual Deduction proofs are contained in set.mm. Examples of Virtual Deduction proofs in mmj2 Proof Worksheet .txt format are included in the completeusersproof download. http://us.metamath.org/other/completeusersproof/suctrvd.html, http://us.metamath.org/other/completeusersproof/sineq0altvd.html, http://us.metamath.org/other/completeusersproof/iunconlem2vd.html, http://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html, and http://us.metamath.org/other/completeusersproof/chordthmaltvd.html are examples of Virtual Deduction proofs. Generally, proving using Virtual Deduction and completeusersproof reduces the amount of Metamath-specific knowledge required by the User. Often, no knowledge of the specific theorems and deductions in set.mm is required to write some of the subproofs of a Virtual Deduction proof. Often, no knowledge of the Metamath-specific names of reference theorems and deductions in set.mm is required for writing some of the subproofs of a User's Proof. Often, the User may write subproofs of a proof using theorems or deductions commonly used in mathematics and correctly assume that some form of each is contained in set.mm and that completeusersproof will automatically generate the technical steps necessary to utilize them to complete the subproofs. Often, the fraction of the work which may be considered tedious is reduced and the total amount of work is reduced. | ||||||||||||||||||||||||||||||
Definition | df-vd1 33469 | Definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | in1 33470 | Inference form of df-vd1 33469. Virtual deduction introduction rule of converting the virtual hypothesis of a 1-virtual hypothesis virtual deduction into an antecedent. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | iin1 33471 | in1 33470 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd1ir 33472 | Inference form of df-vd1 33469 with the virtual deduction as the assertion. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | idn1 33473 | Virtual deduction identity rule which is id 22 with virtual deduction symbols. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd1imp 33474 | Left-to-right part of definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd1impr 33475 | Right-to-left part of definition of virtual deduction. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Syntax | wvd2 33476 | Syntax for a 2-hypothesis virtual deduction. (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Definition | df-vd2 33477 | Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd2 33478 | Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Syntax | wvhc2 33479 | Syntax for a 2-element virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Definition | df-vhc2 33480 | Definition of a 2-element virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd2an 33481 | Definition of a 2-hypothesis virtual deduction in vd conjunction form. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd2ani 33482 | Inference form of dfvd2an 33481. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd2anir 33483 | Right-to-left inference form of dfvd2an 33481. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd2i 33484 | Inference form of dfvd2 33478. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd2ir 33485 | Right-to-left inference form of dfvd2 33478. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Syntax | wvd3 33486 | Syntax for a 3-hypothesis virtual deduction. (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Syntax | wvhc3 33487 | Syntax for a 3-element virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Definition | df-vhc3 33488 | Definition of a 3-element virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Definition | df-vd3 33489 | Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd3 33490 | Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd3i 33491 | Inference form of dfvd3 33490. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd3ir 33492 | Right-to-left inference form of dfvd3 33490. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd3an 33493 | Definition of a 3-hypothesis virtual deduction in vd conjunction form. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd3ani 33494 | Inference form of dfvd3an 33493. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Theorem | dfvd3anir 33495 | Right-to-left inference form of dfvd3an 33493. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Syntax | wvhc4 33496 | Syntax for a 4-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Syntax | wvhc5 33497 | Syntax for a 5-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Syntax | wvhc6 33498 | Syntax for a 6-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Syntax | wvhc7 33499 | Syntax for a 7-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
Syntax | wvhc8 33500 | Syntax for an 8-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
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