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PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY

Logic can be defined as the "study of the principles of correct reasoning" (Merrilee H. Salmon's 1991 "Informal Reasoning and Informal Logic" in Informal Reasoning and Education) or as "a formal system using symbolic techniques and mathematical methods to establish truth-values" (the Oxford English Dictionary).

This section formally defines the logic system we will use. In particular, it defines symbols for declaring truthful statements, along with rules for deriving truthful statements from other truthful statements. The system defined here is classical first-order logic with equality (the most common logic system used by mathematicians).

We begin with a few housekeeping items in pre-logic, and then introduce propositional calculus (both its axioms and important theorems that can be derived from them). Propositional calculus deals with general truths about well-formed formulas (wffs) regardless of how they are constructed. This is followed by proofs that other axiomatizations of classical propositional calculus can be derived from the axioms we have chosen to use.

We then define predicate calculus, which adds additional symbols and rules useful for discussing objects (beyond simply true or false). In particular, it introduces the symbols = ("equals"), ("is a member of"), and ("for all"). The first two are called "predicates." A predicate specifies a true or false relationship between its two arguments.

1.1  Pre-logic

This section includes a few "housekeeping" mechanisms before we begin defining the basics of logic.

1.1.1  Inferences for assisting proof development

The inference rules in this section will normally never appear in a completed proof. They can be ignored if you are using this database to assist learning logic - please start with the statement wn 3 instead.

Theorema1ii 1 (Note: This inference rule and the next one, idi 2, will normally never appear in a completed proof. They can be ignored if you are using this database to assist learning logic - please start with the statement wn 3 instead.)

This is a technical inference to assist proof development. It provides a temporary way to add an independent subproof to a proof under development, for later assignment to a normal proof step.

The metamath program's Proof Assistant requires proofs to be developed backwards from the conclusion with no gaps, and it has no mechanism that lets the user to work on isolated subproofs. This inference provides a workaround for this limitation. It can be inserted at any point in a proof to allow an independent subproof to be developed on the side, for later use as part of the final proof.

Instructions: (1) Assign this inference to any unknown step in the proof. Typically, the last unknown step is the most convenient, since 'assign last' can be used. This step will be replicated in hypothesis a1ii.1, from where the development of the main proof can continue. (2) Develop the independent subproof backwards from hypothesis a1ii.2. If desired, use a 'let' command to pre-assign the conclusion of the independent subproof to a1ii.2. (3) After the independent subproof is complete, use 'improve all' to assign it automatically to an unknown step in the main proof that matches it. (4) After the entire proof is complete, use 'minimize *' to clean up (discard) all a1ii 1 references automatically.

This inference was originally designed to assist importing partially completed Proof Worksheets from the mmj2 Proof Assistant GUI, but it can also be useful on its own. Interestingly, no axioms are required for its proof. It is the inference associated with a1i 11. (Contributed by NM, 7-Feb-2006.)

𝜑    &   𝜓       𝜑

Theoremidi 2 This inference, which requires no axioms for its proof, is useful as a copy-paste mechanism during proof development in mmj2. It is normally not referenced in the final version of a proof, since it is always redundant and can be removed using the 'minimize *' command in the metamath program's Proof Assistant. It is the inference associated with id 22. (Contributed by Alan Sare, 31-Dec-2011.)
𝜑       𝜑

1.2  Propositional calculus

Propositional calculus deals with general truths about well-formed formulas (wffs) regardless of how they are constructed. The simplest propositional truth is (𝜑𝜑), which can be read "if something is true, then it is true" - rather trivial and obvious, but nonetheless it must be proved from the axioms (see theorem id 22).

Our system of propositional calculus consists of three basic axioms and another axiom that defines the modus-ponens inference rule. It is attributed to Jan Lukasiewicz (pronounced woo-kah-SHAY-vitch) and was popularized by Alonzo Church, who called it system P2. (Thanks to Ted Ulrich for this information.) These axioms are ax-1 6, ax-2 7, ax-3 8, and (for modus ponens) ax-mp 5. Some closely followed texts include [Margaris] for the axioms and [WhiteheadRussell] for the theorems.

The propositional calculus used here is the classical system widely used by mathematicians. In particular, this logic system accepts the "law of the excluded middle" as proven in exmid 430, which says that a logical statement is either true or not true. This is an essential distinction of classical logic and is not a theorem of intuitionistic logic.

All 194 axioms, definitions, and theorems for propositional calculus in Principia Mathematica (specifically *1.2 through *5.75) are axioms or formally proven. See the Bibliographic Cross-References at mmbiblio.html for a complete cross-reference from sources used to its formalization in the Metamath Proof Explorer.

1.2.1  Recursively define primitive wffs for propositional calculus

Syntaxwn 3 If 𝜑 is a wff, so is ¬ 𝜑 or "not 𝜑." Part of the recursive definition of a wff (well-formed formula). In classical logic (which is our logic), a wff is interpreted as either true or false. So if 𝜑 is true, then ¬ 𝜑 is false; if 𝜑 is false, then ¬ 𝜑 is true. Traditionally, Greek letters are used to represent wffs, and we follow this convention. In propositional calculus, we define only wffs built up from other wffs, i.e. there is no starting or "atomic" wff. Later, in predicate calculus, we will extend the basic wff definition by including atomic wffs (weq 1861 and wel 1978).
wff ¬ 𝜑

Syntaxwi 4 If 𝜑 and 𝜓 are wff's, so is (𝜑𝜓) or "𝜑 implies 𝜓." Part of the recursive definition of a wff. The resulting wff is (interpreted as) false when 𝜑 is true and 𝜓 is false; it is true otherwise. Think of the truth table for an OR gate with input 𝜑 connected through an inverter. After we define the axioms of propositional calculus (ax-1 6, ax-2 7, ax-3 8, and ax-mp 5), the biconditional (df-bi 196), the constant true (df-tru 1478), and the constant false (df-fal 1481), we will be able to prove these truth table values: ((⊤ → ⊤) ↔ ⊤) (truimtru 1505), ((⊤ → ⊥) ↔ ⊥) (truimfal 1506), ((⊥ → ⊤) ↔ ⊤) (falimtru 1507), and ((⊥ → ⊥) ↔ ⊤) (falimfal 1508). These have straightforward meanings, for example, ((⊤ → ⊤) ↔ ⊤) just means "the value of (⊤ → ⊤) is ".

The left-hand wff is called the antecedent, and the right-hand wff is called the consequent. In the case of (𝜑 → (𝜓𝜒)), the middle 𝜓 may be informally called either an antecedent or part of the consequent depending on context. Contrast with (df-bi 196), (df-an 385), and (df-or 384).

This is called "material implication" and the arrow is usually read as "implies." However, material implication is not identical to the meaning of "implies" in natural language. For example, the word "implies" may suggest a causal relationship in natural language. Material implication does not require any causal relationship. Also, note that in material implication, if the consequent is true then the wff is always true (even if the antecedent is false). Thus, if "implies" means material implication, it is true that "if the moon is made of green cheese that implies that 5=5" (because 5=5). Similarly, if the antecedent is false, the wff is always true. Thus, it is true that, "if the moon made of green cheese that implies that 5=7" (because the moon is not actually made of green cheese). A contradiction implies anything (pm2.21i 115). In short, material implication has a very specific technical definition, and misunderstandings of it are sometimes called "paradoxes of logical implication."

wff (𝜑𝜓)

1.2.2  The axioms of propositional calculus

Propositional calculus (axioms ax-1 6 through ax-3 8 and rule ax-mp 5) can be thought of as asserting formulas that are universally "true" when their variables are replaced by any combination of "true" and "false." Propositional calculus was first formalized by Frege in 1879, using as his axioms (in addition to rule ax-mp 5) the wffs ax-1 6, ax-2 7, pm2.04 88, con3 148, notnot 135, and notnotr 124. Around 1930, Lukasiewicz simplified the system by eliminating the third (which follows from the first two, as you can see by looking at the proof of pm2.04 88) and replacing the last three with our ax-3 8. (Thanks to Ted Ulrich for this information.)

The theorems of propositional calculus are also called tautologies. Tautologies can be proved very simply using truth tables, based on the true/false interpretation of propositional calculus. To do this, we assign all possible combinations of true and false to the wff variables and verify that the result (using the rules described in wi 4 and wn 3) always evaluates to true. This is called the semantic approach. Our approach is called the syntactic approach, in which everything is derived from axioms. A metatheorem called the Completeness Theorem for Propositional Calculus shows that the two approaches are equivalent and even provides an algorithm for automatically generating syntactic proofs from a truth table. Those proofs, however, tend to be long, since truth tables grow exponentially with the number of variables, and the much shorter proofs that we show here were found manually.

Axiomax-mp 5 Rule of Modus Ponens. The postulated inference rule of propositional calculus. See e.g. Rule 1 of [Hamilton] p. 73. The rule says, "if 𝜑 is true, and 𝜑 implies 𝜓, then 𝜓 must also be true." This rule is sometimes called "detachment," since it detaches the minor premise from the major premise. "Modus ponens" is short for "modus ponendo ponens," a Latin phrase that means "the mode that by affirming affirms" - remark in [Sanford] p. 39. This rule is similar to the rule of modus tollens mto 187.

Note: In some web page displays such as the Statement List, the symbols "& " and " " informally indicate the relationship between the hypotheses and the assertion (conclusion), abbreviating the English words "and" and "implies." They are not part of the formal language. (Contributed by NM, 30-Sep-1992.)

𝜑    &   (𝜑𝜓)       𝜓

Axiomax-1 6 Axiom Simp. Axiom A1 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. The 3 axioms are also given as Definition 2.1 of [Hamilton] p. 28. This axiom is called Simp or "the principle of simplification" in Principia Mathematica (Theorem *2.02 of [WhiteheadRussell] p. 100) because "it enables us to pass from the joint assertion of 𝜑 and 𝜓 to the assertion of 𝜑 simply." It is Proposition 1 of [Frege1879] p. 26, its first axiom. (Contributed by NM, 30-Sep-1992.)
(𝜑 → (𝜓𝜑))

Axiomax-2 7 Axiom Frege. Axiom A2 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It "distributes" an antecedent over two consequents. This axiom was part of Frege's original system and is known as Frege in the literature; see Proposition 2 of [Frege1879] p. 26. It is also proved as Theorem *2.77 of [WhiteheadRussell] p. 108. The other direction of this axiom also turns out to be true, as demonstrated by pm5.41 378. (Contributed by NM, 30-Sep-1992.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Axiomax-3 8 Axiom Transp. Axiom A3 of [Margaris] p. 49. One of the 3 axioms of propositional calculus. It swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky." This axiom is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103). We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning. (Contributed by NM, 30-Sep-1992.) Use its alias con4 111 instead. (New usage is discouraged.)
((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑))

1.2.3  Logical implication

The results in this section are based on implication only, and avoid ax-3 8, so are intuitionistic. The system { ax-mp 5, ax-1 6, ax-2 7 } axiomatizes what is sometimes called "intuitionistic implicational calculus" or "minimal implicational calculus."

In an implication, the wff before the arrow is called the "antecedent" and the wff after the arrow is called the "consequent."

Theoremmp2 9 A double modus ponens inference. (Contributed by NM, 5-Apr-1994.)
𝜑    &   𝜓    &   (𝜑 → (𝜓𝜒))       𝜒

Theoremmp2b 10 A double modus ponens inference. (Contributed by Mario Carneiro, 24-Jan-2013.)
𝜑    &   (𝜑𝜓)    &   (𝜓𝜒)       𝜒

Theorema1i 11 Inference introducing an antecedent. Inference associated with ax-1 6. Its associated inference is a1ii 1. See conventions 26650 for a definition of "associated inference". (Contributed by NM, 29-Dec-1992.)
𝜑       (𝜓𝜑)

Theorem2a1i 12 Inference introducing two antecedents. Two applications of a1i 11. Inference associated with 2a1 28. (Contributed by Jeff Hankins, 4-Aug-2009.)
𝜑       (𝜓 → (𝜒𝜑))

Theoremmp1i 13 Inference detaching an antecedent and introducing a new one. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝜑    &   (𝜑𝜓)       (𝜒𝜓)

Theorema2i 14 Inference distributing an antecedent. Inference associated with ax-2 7. Its associated inference is mpd 15. (Contributed by NM, 29-Dec-1992.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → (𝜑𝜒))

Theoremmpd 15 A modus ponens deduction. A translation of natural deduction rule E ( elimination), see natded 26652. Deduction form of ax-mp 5. Inference associated with a2i 14. (Contributed by NM, 29-Dec-1992.)
(𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremimim2i 16 Inference adding common antecedents in an implication. Inference associated with imim2 56. Its associated inference is syl 17. (Contributed by NM, 28-Dec-1992.)
(𝜑𝜓)       ((𝜒𝜑) → (𝜒𝜓))

Theoremsyl 17 An inference version of the transitive laws for implication imim2 56 and imim1 81 (and imim1i 61 and imim2i 16), which Russell and Whitehead call "the principle of the syllogism...because...the syllogism in Barbara is derived from them" (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Some authors call this law a "hypothetical syllogism." Its associated inference is mp2b 10.

(A bit of trivia: this is the most commonly referenced assertion in our database (13449 times as of 22-Jul-2021). In second place is eqid 2610 (9597 times), followed by adantr 480 (8861 times), syl2anc 691 (7421 times), adantl 481 (6403 times), and simpr 476 (5829 times). The Metamath program command 'show usage' shows the number of references.)

(Contributed by NM, 30-Sep-1992.) (Proof shortened by Mel L. O'Cat, 20-Oct-2011.) (Proof shortened by Wolf Lammen, 26-Jul-2012.)

(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theorem3syl 18 Inference chaining two syllogisms syl 17. Inference associated with imim12i 60. (Contributed by NM, 28-Dec-1992.)
(𝜑𝜓)    &   (𝜓𝜒)    &   (𝜒𝜃)       (𝜑𝜃)

Theorem4syl 19 Inference chaining three syllogisms syl 17. (Contributed by BJ, 14-Jul-2018.) The use of this theorem is marked "discouraged" because it can cause the "minimize" command to have very long run times. However, feel free to use "minimize 4syl /override" if you wish. Remember to update the Travis "discouraged" file if it gets used. (New usage is discouraged.)
(𝜑𝜓)    &   (𝜓𝜒)    &   (𝜒𝜃)    &   (𝜃𝜏)       (𝜑𝜏)

Theoremmpi 20 A nested modus ponens inference. Inference associated with com12 32. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Stefan Allan, 20-Mar-2006.)
𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremmpisyl 21 A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.)
(𝜑𝜓)    &   𝜒    &   (𝜓 → (𝜒𝜃))       (𝜑𝜃)

Theoremid 22 Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. For another version of the proof directly from axioms, see idALT 23. Its associated inference, idi 2, requires no axioms for its proof, contrary to id 22. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Stefan Allan, 20-Mar-2006.)
(𝜑𝜑)

TheoremidALT 23 Alternate proof of id 22. This version is proved directly from the axioms for demonstration purposes. This proof is a popular example in the literature and is identical, step for step, to the proofs of Theorem 1 of [Margaris] p. 51, Example 2.7(a) of [Hamilton] p. 31, Lemma 10.3 of [BellMachover] p. 36, and Lemma 1.8 of [Mendelson] p. 36. It is also "Our first proof" in Hirst and Hirst's A Primer for Logic and Proof p. 17 (PDF p. 23) at http://www.appstate.edu/~hirstjl/primer/hirst.pdf. Note that the second occurrence of 𝜑 in Steps 1 to 4 and the sixth in Step 3 may simultaneously be replaced by any wff 𝜓, which may ease the understanding of the proof. For a shorter version of the proof that takes advantage of previously proved theorems, see id 22. (Contributed by NM, 30-Sep-1992.) (Proof modification is discouraged.) Use id 22 instead. (New usage is discouraged.)
(𝜑𝜑)

Theoremidd 24 Principle of identity id 22 with antecedent. (Contributed by NM, 26-Nov-1995.)
(𝜑 → (𝜓𝜓))

Theorema1d 25 Deduction introducing an embedded antecedent. Deduction form of ax-1 6 and a1i 11. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Stefan Allan, 20-Mar-2006.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))

Theorem2a1d 26 Deduction introducing two antecedents. Two applications of a1d 25. Deduction associated with 2a1 28 and 2a1i 12. (Contributed by BJ, 10-Aug-2020.)
(𝜑𝜓)       (𝜑 → (𝜒 → (𝜃𝜓)))

Theorema1i13 27 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
(𝜓𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))

Theorem2a1 28 A double form of ax-1 6. Its associated inference is 2a1i 12. Its associated deduction is 2a1d 26. (Contributed by BJ, 10-Aug-2020.) (Proof shortened by Wolf Lammen, 1-Sep-2020.)
(𝜑 → (𝜓 → (𝜒𝜑)))

Theorema2d 29 Deduction distributing an embedded antecedent. Deduction form of ax-2 7. (Contributed by NM, 23-Jun-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))

Theoremsylcom 30 Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜓 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))

Theoremsyl5com 31 Syllogism inference with commuted antecedents. (Contributed by NM, 24-May-2005.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜑 → (𝜒𝜃))

Theoremcom12 32 Inference that swaps (commutes) antecedents in an implication. Inference associated with pm2.04 88. Its associated inference is mpi 20. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)
(𝜑 → (𝜓𝜒))       (𝜓 → (𝜑𝜒))

Theoremsyl5 33 A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 25-May-2013.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))

Theoremsyl6 34 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Wolf Lammen, 30-Jul-2012.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))

Theoremsyl56 35 Combine syl5 33 and syl6 34. (Contributed by NM, 14-Nov-2013.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))    &   (𝜃𝜏)       (𝜒 → (𝜑𝜏))

Theoremsyl6com 36 Syllogism inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜓 → (𝜑𝜃))

Theoremmpcom 37 Modus ponens inference with commutation of antecedents. (Contributed by NM, 17-Mar-1996.)
(𝜓𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜓𝜒)

Theoremsyli 38 Syllogism inference with common nested antecedent. (Contributed by NM, 4-Nov-2004.)
(𝜓 → (𝜑𝜒))    &   (𝜒 → (𝜑𝜃))       (𝜓 → (𝜑𝜃))

Theoremsyl2im 39 Replace two antecedents. Implication-only version of syl2an 493. (Contributed by Wolf Lammen, 14-May-2013.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜓 → (𝜃𝜏))       (𝜑 → (𝜒𝜏))

Theoremsyl2imc 40 A commuted version of syl2im 39. Implication-only version of syl2anr 494. (Contributed by BJ, 20-Oct-2021.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜓 → (𝜃𝜏))       (𝜒 → (𝜑𝜏))

Theorempm2.27 41 This theorem, called "Assertion," can be thought of as closed form of modus ponens ax-mp 5. Theorem *2.27 of [WhiteheadRussell] p. 104. (Contributed by NM, 15-Jul-1993.)
(𝜑 → ((𝜑𝜓) → 𝜓))

Theoremmpdd 42 A nested modus ponens deduction. Double deduction associated with ax-mp 5. Deduction associated with mpd 15. (Contributed by NM, 12-Dec-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓𝜃))

Theoremmpid 43 A nested modus ponens deduction. Deduction associated with mpi 20. (Contributed by NM, 14-Dec-2004.)
(𝜑𝜒)    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓𝜃))

Theoremmpdi 44 A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.) (Proof shortened by Mel L. O'Cat, 15-Jan-2008.)
(𝜓𝜒)    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓𝜃))

Theoremmpii 45 A doubly nested modus ponens inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.)
𝜒    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓𝜃))

Theoremsyld 46 Syllogism deduction. Deduction associated with syl 17. See conventions 26650 for the meaning of "associated deduction" or "deduction form". (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mel L. O'Cat, 19-Feb-2008.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓𝜃))

Theoremmp2d 47 A double modus ponens deduction. Deduction associated with mp2 9. (Contributed by NM, 23-May-2013.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑𝜃)

Theorema1dd 48 Double deduction introducing an antecedent. Deduction associated with a1d 25. Double deduction associated with ax-1 6 and a1i 11. (Contributed by NM, 17-Dec-2004.) (Proof shortened by Mel L. O'Cat, 15-Jan-2008.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜃𝜒)))

Theorem2a1dd 49 Double deduction introducing two antecedents. Two applications of 2a1dd 49. Deduction associated with 2a1d 26. Double deduction associated with 2a1 28 and 2a1i 12. (Contributed by Jeff Hankins, 5-Aug-2009.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜃 → (𝜏𝜒))))

Theorempm2.43i 50 Inference absorbing redundant antecedent. Inference associated with pm2.43 54. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)
(𝜑 → (𝜑𝜓))       (𝜑𝜓)

Theorempm2.43d 51 Deduction absorbing redundant antecedent. Deduction associated with pm2.43 54 and pm2.43i 50. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)
(𝜑 → (𝜓 → (𝜓𝜒)))       (𝜑 → (𝜓𝜒))

Theorempm2.43a 52 Inference absorbing redundant antecedent. (Contributed by NM, 7-Nov-1995.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)
(𝜓 → (𝜑 → (𝜓𝜒)))       (𝜓 → (𝜑𝜒))

Theorempm2.43b 53 Inference absorbing redundant antecedent. (Contributed by NM, 31-Oct-1995.)
(𝜓 → (𝜑 → (𝜓𝜒)))       (𝜑 → (𝜓𝜒))

Theorempm2.43 54 Absorption of redundant antecedent. Also called the "Contraction" or "Hilbert" axiom. Theorem *2.43 of [WhiteheadRussell] p. 106. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Mel L. O'Cat, 15-Aug-2004.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))

Theoremimim2d 55 Deduction adding nested antecedents. Deduction associated with imim2 56 and imim2i 16. (Contributed by NM, 10-Jan-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) → (𝜃𝜒)))

Theoremimim2 56 A closed form of syllogism (see syl 17). Theorem *2.05 of [WhiteheadRussell] p. 100. Its associated inference is imim2i 16. Its associated deduction is imim2d 55. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 6-Sep-2012.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))

Theoremembantd 57 Deduction embedding an antecedent. (Contributed by Wolf Lammen, 4-Oct-2013.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → 𝜃))

Theorem3syld 58 Triple syllogism deduction. Deduction associated with 3syld 58. (Contributed by Jeff Hankins, 4-Aug-2009.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theoremsylsyld 59 A double syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   (𝜓 → (𝜃𝜏))       (𝜑 → (𝜒𝜏))

Theoremimim12i 60 Inference joining two implications. Inference associated with imim12 103. Its associated inference is 3syl 18. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Mel L. O'Cat, 29-Oct-2011.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜓𝜒) → (𝜑𝜃))

Theoremimim1i 61 Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. Inference associated with imim1 81. Its associated inference is syl 17. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)
(𝜑𝜓)       ((𝜓𝜒) → (𝜑𝜒))

Theoremimim3i 62 Inference adding three nested antecedents. (Contributed by NM, 19-Dec-2006.)
(𝜑 → (𝜓𝜒))       ((𝜃𝜑) → ((𝜃𝜓) → (𝜃𝜒)))

Theoremsylc 63 A syllogism inference combined with contraction. (Contributed by NM, 4-May-1994.) (Revised by NM, 13-Jul-2013.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜓 → (𝜒𝜃))       (𝜑𝜃)

Theoremsyl3c 64 A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (𝜑𝜏)

Theoremsyl6mpi 65 A syllogism inference. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.)
(𝜑 → (𝜓𝜒))    &   𝜃    &   (𝜒 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theoremmpsyl 66 Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)
𝜑    &   (𝜓𝜒)    &   (𝜑 → (𝜒𝜃))       (𝜓𝜃)

Theoremmpsylsyld 67 Modus ponens combined with a double syllogism inference. (Contributed by Alan Sare, 22-Jul-2012.)
𝜑    &   (𝜓 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜏))       (𝜓 → (𝜒𝜏))

Theoremsyl6c 68 Inference combining syl6 34 with contraction. (Contributed by Alan Sare, 2-May-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜒 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theoremsyl6ci 69 A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)    &   (𝜒 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theoremsyldd 70 Nested syllogism deduction. Deduction associated with syld 46. Double deduction associated with syl 17. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜃𝜏)))       (𝜑 → (𝜓 → (𝜒𝜏)))

Theoremsyl5d 71 A nested syllogism deduction. Deduction associated with syl5 33. (Contributed by NM, 14-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃 → (𝜒𝜏)))       (𝜑 → (𝜃 → (𝜓𝜏)))

Theoremsyl7 72 A syllogism rule of inference. The first premise is used to replace the third antecedent of the second premise. (Contributed by NM, 12-Jan-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
(𝜑𝜓)    &   (𝜒 → (𝜃 → (𝜓𝜏)))       (𝜒 → (𝜃 → (𝜑𝜏)))

Theoremsyl6d 73 A nested syllogism deduction. Deduction associated with syl6 34. (Contributed by NM, 11-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜃𝜏))       (𝜑 → (𝜓 → (𝜒𝜏)))

Theoremsyl8 74 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜃𝜏)       (𝜑 → (𝜓 → (𝜒𝜏)))

Theoremsyl9 75 A nested syllogism inference with different antecedents. (Contributed by NM, 13-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       (𝜑 → (𝜃 → (𝜓𝜏)))

Theoremsyl9r 76 A nested syllogism inference with different antecedents. (Contributed by NM, 14-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       (𝜃 → (𝜑 → (𝜓𝜏)))

Theoremsyl10 77 A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜃𝜏)))    &   (𝜒 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜃𝜂)))

Theorema1ddd 78 Triple deduction introducing an antecedent to a wff. Deduction associated with a1dd 48. Double deduction associated with a1d 25. Triple deduction associated with ax-1 6 and a1i 11. (Contributed by Jeff Hankins, 4-Aug-2009.)
(𝜑 → (𝜓 → (𝜒𝜏)))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremimim12d 79 Deduction combining antecedents and consequents. Deduction associated with imim12 103 and imim12i 60. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mel L. O'Cat, 30-Oct-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜒𝜃) → (𝜓𝜏)))

Theoremimim1d 80 Deduction adding nested consequents. Deduction associated with imim1 81 and imim1i 61. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜒𝜃) → (𝜓𝜃)))

Theoremimim1 81 A closed form of syllogism (see syl 17). Theorem *2.06 of [WhiteheadRussell] p. 100. Its associated inference is imim1i 61. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 25-May-2013.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theorempm2.83 82 Theorem *2.83 of [WhiteheadRussell] p. 108. Closed form of syld 46. (Contributed by NM, 3-Jan-2005.)
((𝜑 → (𝜓𝜒)) → ((𝜑 → (𝜒𝜃)) → (𝜑 → (𝜓𝜃))))

Theorempeirceroll 83 Over minimal implicational calculus, Peirce's axiom peirce 192 implies an axiom sometimes called "Roll", (((𝜑𝜓) → 𝜒) → ((𝜒𝜑) → 𝜑)), of which looinv 193 is a special instance. The converse also holds: substitute (𝜑𝜓) for 𝜒 in Roll and use id 22 and ax-mp 5. (Contributed by BJ, 15-Jun-2021.)
((((𝜑𝜓) → 𝜑) → 𝜑) → (((𝜑𝜓) → 𝜒) → ((𝜒𝜑) → 𝜑)))

Theoremcom23 84 Commutation of antecedents. Swap 2nd and 3rd. Deduction associated with com12 32. (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜒 → (𝜓𝜃)))

Theoremcom3r 85 Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜒 → (𝜑 → (𝜓𝜃)))

Theoremcom13 86 Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜒 → (𝜓 → (𝜑𝜃)))

Theoremcom3l 87 Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜓 → (𝜒 → (𝜑𝜃)))

Theorempm2.04 88 Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. This was the third axiom in Frege's logic system, specifically Proposition 8 of [Frege1879] p. 35. Its associated inference is com12 32. (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

Theoremcom34 89 Commutation of antecedents. Swap 3rd and 4th. Deduction associated with com23 84. (Contributed by NM, 25-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (𝜓 → (𝜃 → (𝜒𝜏))))

Theoremcom4l 90 Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Mel L. O'Cat, 15-Aug-2004.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜓 → (𝜒 → (𝜃 → (𝜑𝜏))))

Theoremcom4t 91 Commutation of antecedents. Rotate twice. (Contributed by NM, 25-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜒 → (𝜃 → (𝜑 → (𝜓𝜏))))

Theoremcom4r 92 Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜃 → (𝜑 → (𝜓 → (𝜒𝜏))))

Theoremcom24 93 Commutation of antecedents. Swap 2nd and 4th. Deduction associated with com13 86. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (𝜃 → (𝜒 → (𝜓𝜏))))

Theoremcom14 94 Commutation of antecedents. Swap 1st and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜃 → (𝜓 → (𝜒 → (𝜑𝜏))))

Theoremcom45 95 Commutation of antecedents. Swap 4th and 5th. Deduction associated with com34 89. (Contributed by Jeff Hankins, 28-Jun-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜓 → (𝜒 → (𝜏 → (𝜃𝜂)))))

Theoremcom35 96 Commutation of antecedents. Swap 3rd and 5th. Deduction associated with com24 93. (Contributed by Jeff Hankins, 28-Jun-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜓 → (𝜏 → (𝜃 → (𝜒𝜂)))))

Theoremcom25 97 Commutation of antecedents. Swap 2nd and 5th. Deduction associated with com14 94. (Contributed by Jeff Hankins, 28-Jun-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜏 → (𝜒 → (𝜃 → (𝜓𝜂)))))

Theoremcom5l 98 Commutation of antecedents. Rotate left. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜓 → (𝜒 → (𝜃 → (𝜏 → (𝜑𝜂)))))

Theoremcom15 99 Commutation of antecedents. Swap 1st and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜏 → (𝜓 → (𝜒 → (𝜃 → (𝜑𝜂)))))

Theoremcom52l 100 Commutation of antecedents. Rotate left twice. (Contributed by Jeff Hankins, 28-Jun-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜒 → (𝜃 → (𝜏 → (𝜑 → (𝜓𝜂)))))

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