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Theorem List for Intuitionistic Logic Explorer - 1-100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
PART 1  FIRST ORDER LOGIC WITH EQUALITY
 
1.1  Pre-logic
 
1.1.1  Inferences for assisting proof development
 
Theoremdummylink 1 (Note: This inference rule and the next one, idi 2, will normally never appear in a completed proof. It 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 dummylink.1, from where the development of the main proof can continue. (2) Develop the independent subproof backwards from hypothesis dummylink.2. If desired, use a 'let' command to pre-assign the conclusion of the independent subproof to dummylink.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 dummylink 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. (Contributed by NM, 7-Feb-2006.)

   &       =>   
 
Theoremidi 2 Inference form of id 17. This inference rule, 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. (Contributed by Alan Sare, 31-Dec-2011.)
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1.2  Propositional calculus
 
1.2.1  Recursively define primitive wffs for propositional calculus
 
Syntaxwn 3 If is a wff, so is or "not ." Part of the recursive definition of a wff (well-formed formula). 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 1323 and wel 1325).
 
Syntaxwi 4 If and are wff's, so is or " implies ." Part of the recursive definition of a wff. The left-hand wff is called the antecedent, and the right-hand wff is called the consequent. In the case of , the middle may be informally called either an antecedent or part of the consequent depending on context.
 
1.2.2  The axioms of propositional calculus
 
Axiomax-1 5 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 and to the assertion of simply."

General remarks: Propositional calculus (axioms ax-1 5 through ax-3 1952 and rule ax-mp 7) 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 7) the wffs ax-1 5, ax-2 6, pm2.04 74, con3 551, notnot2 1962, and notnot1 541. 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 74) and replacing the last three with our ax-3 1952. (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. (Contributed by NM, 5-Aug-1993.)

 
Axiomax-2 6 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. 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 pm5.41 238. (Contributed by NM, 5-Aug-1993.)
 
Axiomax-mp 7 Rule of Modus Ponens. The postulated inference rule of propositional calculus. See e.g. Rule 1 of [Hamilton] p. 73. The rule says, "if is true, and implies , then must also be true." This rule is sometimes called "detachment," since it detaches the minor premise from the major premise.

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, 5-Aug-1993.)

   &       =>   
 
1.2.3  Logical implication

The results in this section are based on implication only, and avoid ax-3 1952. In an implication, the wff before the arrow is called the "antecedent" and the wff after the arrow is called the "consequent."

We will use the following descriptive terms very loosely: A "closed form" or "tautology" has no $e hypotheses. An "inference" has one or more $e hypotheses. A "deduction" is an inference in which the hypotheses and the conclusion share the same antecedent.

 
Theoremmp2b 8 A double modus ponens inference. (Contributed by Mario Carneiro, 24-Jan-2013.)
   &       &       =>   
 
Theorema1i 9 Inference derived from axiom ax-1 5. See a1d 20 for an explanation of our informal use of the terms "inference" and "deduction." See also the comment in syld 38. (Contributed by NM, 5-Aug-1993.)
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Theorema2i 10 Inference derived from axiom ax-2 6. (Contributed by NM, 5-Aug-1993.)
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Theoremimim2i 11 Inference adding common antecedents in an implication. (Contributed by NM, 5-Aug-1993.)
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Theoremmpd 12 A modus ponens deduction. (Contributed by NM, 5-Aug-1993.)
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Theoremsyl 13 An inference version of the transitive laws for implication imim2 47 and imim1 68, 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." (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 20-Oct-2011.) (Proof shortened by Wolf Lammen, 26-Jul-2012.)
   &       =>   
 
Theoremmpi 14 A nested modus ponens inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Stefan Allan, 20-Mar-2006.)
   &       =>   
 
Theoremmp2 15 A double modus ponens inference. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
   &       &       =>   
 
Theorem3syl 16 Inference chaining two syllogisms. (Contributed by NM, 5-Aug-1993.)
   &       &       =>   
 
Theoremid 17 Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. For another version of the proof directly from axioms, see id1 18. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Stefan Allan, 20-Mar-2006.)
 
Theoremid1 18 Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. 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.mathsci.appstate.edu/~jlh/primer/hirst.pdf. For a shorter version of the proof that takes advantage of previously proved theorems, see id 17. (Contributed by NM, 5-Aug-1993.)
 
Theoremidd 19 Principle of identity with antecedent. (Contributed by NM, 26-Nov-1995.)
 
Theorema1d 20 Deduction introducing an embedded antecedent. (The proof was revised by Stefan Allan, 20-Mar-2006.)

Naming convention: We often call a theorem a "deduction" and suffix its label with "d" whenever the hypotheses and conclusion are each prefixed with the same antecedent. This allows us to use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem would be used; here would be replaced with a conjunction (wa 95) of the hypotheses of the would-be deduction. By contrast, we tend to call the simpler version with no common antecedent an "inference" and suffix its label with "i"; compare theorem a1i 9. Finally, a "theorem" would be the form with no hypotheses; in this case the "theorem" form would be the original axiom ax-1 5. We usually show the theorem form without a suffix on its label (e.g. pm2.43 45 vs. pm2.43i 41 vs. pm2.43d 42). (Contributed by NM, 5-Aug-1993.) (Revised by NM, 20-Mar-2006.)

   =>   
 
Theorema2d 21 Deduction distributing an embedded antecedent. (Contributed by NM, 23-Jun-1994.)
   =>   
 
Theorema1ii 22 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
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Theoremsylcom 23 Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.)
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Theoremsyl5com 24 Syllogism inference with commuted antecedents. (Contributed by NM, 24-May-2005.)
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Theoremcom12 25 Inference that swaps (commutes) antecedents in an implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)
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Theoremsyl5 26 A syllogism rule of inference. The second premise is used to replace the second antecedent of the first premise. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-May-2013.)
   &       =>   
 
Theoremsyl6 27 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jul-2012.)
   &       =>   
 
Theoremsyl56 28 Combine syl5 26 and syl6 27. (Contributed by NM, 14-Nov-2013.)
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Theoremsyl6com 29 Syllogism inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
   &       =>   
 
Theoremmpcom 30 Modus ponens inference with commutation of antecedents. (Contributed by NM, 17-Mar-1996.)
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Theoremsyli 31 Syllogism inference with common nested antecedent. (Contributed by NM, 4-Nov-2004.)
   &       =>   
 
Theoremsyl2im 32 Replace two antecedents. Implication-only version of syl2an 271. (Contributed by Wolf Lammen, 14-May-2013.)
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Theorempm2.27 33 This theorem, called "Assertion," can be thought of as closed form of modus ponens ax-mp 7. Theorem *2.27 of [WhiteheadRussell] p. 104. (Contributed by NM, 5-Aug-1993.)
 
Theoremmpdd 34 A nested modus ponens deduction. (Contributed by NM, 12-Dec-2004.)
   &       =>   
 
Theoremmpid 35 A nested modus ponens deduction. (Contributed by NM, 14-Dec-2004.)
   &       =>   
 
Theoremmpdi 36 A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.) (Proof shortened by O'Cat, 15-Jan-2008.)
   &       =>   
 
Theoremmpii 37 A doubly nested modus ponens inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.)
   &       =>   
 
Theoremsyld 38 Syllogism deduction.

Notice that syld 38 has the same form as syl 13 with added in front of each hypothesis and conclusion. When all theorems referenced in a proof are converted in this way, we can replace with a hypothesis of the proof, allowing the hypothesis to be eliminated with id 17 and become an antecedent. The Deduction Theorem for propositional calculus, e.g. Theorem 3 in [Margaris] p. 56, tells us that this procedure is always possible. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 19-Feb-2008.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)

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Theoremmp2d 39 A double modus ponens deduction. (Contributed by NM, 23-May-2013.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
   &       &       =>   
 
Theorema1dd 40 Deduction introducing a nested embedded antecedent. (Contributed by NM, 17-Dec-2004.) (Proof shortened by O'Cat, 15-Jan-2008.)
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Theorempm2.43i 41 Inference absorbing redundant antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
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Theorempm2.43d 42 Deduction absorbing redundant antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
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Theorempm2.43a 43 Inference absorbing redundant antecedent. (Contributed by NM, 7-Nov-1995.) (Proof shortened by O'Cat, 28-Nov-2008.)
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Theorempm2.43b 44 Inference absorbing redundant antecedent. (Contributed by NM, 31-Oct-1995.)
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Theorempm2.43 45 Absorption of redundant antecedent. Also called the "Contraction" or "Hilbert" axiom. Theorem *2.43 of [WhiteheadRussell] p. 106. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 15-Aug-2004.)
 
Theoremimim2d 46 Deduction adding nested antecedents. (Contributed by NM, 5-Aug-1993.)
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Theoremimim2 47 A closed form of syllogism (see syl 13). Theorem *2.05 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 6-Sep-2012.)
 
Theoremembantd 48 Deduction embedding an antecedent. (Contributed by Wolf Lammen, 4-Oct-2013.)
   &       =>   
 
Theorem3syld 49 Triple syllogism deduction. (Contributed by Jeff Hankins, 4-Aug-2009.)
   &       &       =>   
 
Theoremsylsyld 50 Virtual deduction rule. (Contributed by Alan Sare, 20-Apr-2011.)
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Theoremimim12i 51 Inference joining two implications. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 29-Oct-2011.)
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Theoremimim1i 52 Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)
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Theoremimim3i 53 Inference adding three nested antecedents. (Contributed by NM, 19-Dec-2006.)
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Theoremsylc 54 A syllogism inference combined with contraction. (Contributed by NM, 4-May-1994.) (Revised by NM, 13-Jul-2013.)
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Theoremsyl3c 55 A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.)
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Theoremsyl6mpi 56 syl6 27 combined with mpi 14. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.)
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Theoremmpsyl 57 Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)
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Theoremsyl6c 58 Inference combining syl6 27 with contraction. (Contributed by Alan Sare, 2-May-2011.)
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Theoremsyldd 59 Nested syllogism deduction. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.)
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Theoremsyl5d 60 A nested syllogism deduction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by O'Cat, 2-Feb-2006.)
   &       =>   
 
Theoremsyl7 61 A syllogism rule of inference. The second premise is used to replace the third antecedent of the first premise. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)
   &       =>   
 
Theoremsyl6d 62 A nested syllogism deduction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by O'Cat, 2-Feb-2006.) (Revised by NM, 3-Feb-2006.)
   &       =>   
 
Theoremsyl8 63 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.)
   &       =>   
 
Theoremsyl9 64 A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
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Theoremsyl9r 65 A nested syllogism inference with different antecedents. (Contributed by NM, 5-Aug-1993.)
   &       =>   
 
Theoremimim12d 66 Deduction combining antecedents and consequents. (Contributed by NM, 7-Aug-1994.) (Proof shortened by O'Cat, 30-Oct-2011.)
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Theoremimim1d 67 Deduction adding nested consequents. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)
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Theoremimim1 68 A closed form of syllogism (see syl 13). Theorem *2.06 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-May-2013.)
 
Theorempm2.83 69 Theorem *2.83 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
 
Theoremcom23 70 Commutation of antecedents. Swap 2nd and 3rd. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2012.)
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Theoremcom3r 71 Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)
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Theoremcom13 72 Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
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Theoremcom3l 73 Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
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Theorempm2.04 74 Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Sep-2012.)
 
Theoremcom34 75 Commutation of antecedents. Swap 3rd and 4th. (Contributed by NM, 25-Apr-1994.)
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Theoremcom4l 76 Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by O'Cat, 15-Aug-2004.)
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Theoremcom4t 77 Commutation of antecedents. Rotate twice. (Contributed by NM, 25-Apr-1994.)
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Theoremcom4r 78 Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.)
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Theoremcom24 79 Commutation of antecedents. Swap 2nd and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
   =>   
 
Theoremcom14 80 Commutation of antecedents. Swap 1st and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.)
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Theoremcom45 81 Commutation of antecedents. Swap 4th and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.)
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Theoremcom35 82 Commutation of antecedents. Swap 3rd and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.)
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Theoremcom25 83 Commutation of antecedents. Swap 2nd and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.)
   =>   
 
Theoremcom5l 84 Commutation of antecedents. Rotate left. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)
   =>   
 
Theoremcom15 85 Commutation of antecedents. Swap 1st and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)
   =>   
 
Theoremcom52l 86 Commutation of antecedents. Rotate left twice. (Contributed by Jeff Hankins, 28-Jun-2009.)
   =>   
 
Theoremcom52r 87 Commutation of antecedents. Rotate right twice. (Contributed by Jeff Hankins, 28-Jun-2009.)
   =>   
 
Theoremcom5r 88 Commutation of antecedents. Rotate right. (Contributed by Wolf Lammen, 29-Jul-2012.)
   =>   
 
Theoremjarr 89 Elimination of a nested antecedent as a kind of reversal of inference ja 1960. (Contributed by Wolf Lammen, 9-May-2013.)
 
Theorempm2.86i 90 Inference based on pm2.86 92. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
   =>   
 
Theorempm2.86d 91 Deduction based on pm2.86 92. (Contributed by NM, 29-Jun-1995.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
   =>   
 
Theorempm2.86 92 Converse of axiom ax-2 6. Theorem *2.86 of [WhiteheadRussell] p. 108. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
 
Theoremloolin 93 The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. (Contributed by O'Cat, 12-Aug-2004.)
 
Theoremloowoz 94 An alternate for the Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, due to Barbara Wozniakowska, Reports on Mathematical Logic 10, 129-137 (1978). (Contributed by O'Cat, 8-Aug-2004.)
 
1.2.4  Logical conjunction and logical equivalence
 
Syntaxwa 95 Extend wff definition to include conjunction ('and').
 
Syntaxwb 96 Extend our wff definition to include the biconditional connective.
 
Axiomax-ia1 97 Left 'and' elimination. Axiom 1 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
 
Axiomax-ia2 98 Right 'and' elimination. Axiom 2 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
 
Axiomax-ia3 99 'And' introduction. Axiom 3 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
 
Theoremsimpl 100 Elimination of a conjunct. Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
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