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Theorem List for Metamath Proof Explorer - 1001-1100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremandi 1001 Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒)))
 
Theoremandir 1002 Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))
 
Theoremorddi 1003 Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ (((𝜑𝜒) ∧ (𝜑𝜃)) ∧ ((𝜓𝜒) ∧ (𝜓𝜃))))
 
Theoremanddi 1004 Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ (((𝜑𝜒) ∨ (𝜑𝜃)) ∨ ((𝜓𝜒) ∨ (𝜓𝜃))))
 
Theorempm5.17 1005 Theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)
(((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ (𝜑 ↔ ¬ 𝜓))
 
Theorempm5.15 1006 Theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 15-Oct-2013.)
((𝜑𝜓) ∨ (𝜑 ↔ ¬ 𝜓))
 
Theorempm5.16 1007 Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 17-Oct-2013.)
¬ ((𝜑𝜓) ∧ (𝜑 ↔ ¬ 𝜓))
 
Theoremxor 1008 Two ways to express exclusive disjunction (df-xor 1496). Theorem *5.22 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 22-Jan-2013.)
(¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
 
Theoremnbi2 1009 Two ways to express "exclusive or". (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)
(¬ (𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
 
Theoremxordi 1010 Conjunction distributes over exclusive-or, using ¬ (𝜑𝜓) to express exclusive-or. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. This is not necessarily true in intuitionistic logic, though anxordi 1510 does hold in it. (Contributed by NM, 3-Oct-2008.)
((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
 
Theorempm5.54 1011 Theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 7-Nov-2013.)
(((𝜑𝜓) ↔ 𝜑) ∨ ((𝜑𝜓) ↔ 𝜓))
 
Theorempm5.62 1012 Theorem *5.62 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 21-Jun-2005.)
(((𝜑𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))
 
Theorempm5.63 1013 Theorem *5.63 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 25-Dec-2012.)
((𝜑𝜓) ↔ (𝜑 ∨ (¬ 𝜑𝜓)))
 
Theoremniabn 1014 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
𝜑       𝜓 → ((𝜒𝜓) ↔ ¬ 𝜑))
 
Theoremninba 1015 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
𝜑       𝜓 → (¬ 𝜑 ↔ (𝜒𝜓)))
 
Theorempm4.43 1016 Theorem *4.43 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)
(𝜑 ↔ ((𝜑𝜓) ∧ (𝜑 ∨ ¬ 𝜓)))
 
Theorempm4.82 1017 Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)
 
Theorempm4.83 1018 Theorem *4.83 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (¬ 𝜑𝜓)) ↔ 𝜓)
 
Theorempclem6 1019 Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Nov-2012.)
((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)
 
Theorembigolden 1020 Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)
(((𝜑𝜓) ↔ 𝜑) ↔ (𝜓 ↔ (𝜑𝜓)))
 
Theorempm5.71 1021 Theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.)
((𝜓 → ¬ 𝜒) → (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑𝜒)))
 
Theorempm5.75 1022 Theorem *5.75 of [WhiteheadRussell] p. 126. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.) (Proof shortened by Kyle Wyonch, 12-Feb-2021.)
(((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))
 
Theoremecase2d 1023 Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Dec-2012.)
(𝜑𝜓)    &   (𝜑 → ¬ (𝜓𝜒))    &   (𝜑 → ¬ (𝜓𝜃))    &   (𝜑 → (𝜏 ∨ (𝜒𝜃)))       (𝜑𝜏)
 
Theoremecase3 1024 Inference for elimination by cases. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)
(𝜑𝜒)    &   (𝜓𝜒)    &   (¬ (𝜑𝜓) → 𝜒)       𝜒
 
Theoremecase 1025 Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)
𝜑𝜒)    &   𝜓𝜒)    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoremecase3d 1026 Deduction for elimination by cases. (Contributed by NM, 2-May-1996.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (¬ (𝜓𝜒) → 𝜃))       (𝜑𝜃)
 
Theoremecased 1027 Deduction for elimination by cases. (Contributed by NM, 8-Oct-2012.)
(𝜑 → (¬ 𝜓𝜃))    &   (𝜑 → (¬ 𝜒𝜃))    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑𝜃)
 
Theoremecase3ad 1028 Deduction for elimination by cases. (Contributed by NM, 24-May-2013.)
(𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → ((¬ 𝜓 ∧ ¬ 𝜒) → 𝜃))       (𝜑𝜃)
 
Theoremccase 1029 Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)
((𝜑𝜓) → 𝜏)    &   ((𝜒𝜓) → 𝜏)    &   ((𝜑𝜃) → 𝜏)    &   ((𝜒𝜃) → 𝜏)       (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)
 
Theoremccased 1030 Deduction for combining cases. (Contributed by NM, 9-May-2004.)
(𝜑 → ((𝜓𝜒) → 𝜂))    &   (𝜑 → ((𝜃𝜒) → 𝜂))    &   (𝜑 → ((𝜓𝜏) → 𝜂))    &   (𝜑 → ((𝜃𝜏) → 𝜂))       (𝜑 → (((𝜓𝜃) ∧ (𝜒𝜏)) → 𝜂))
 
Theoremccase2 1031 Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
((𝜑𝜓) → 𝜏)    &   (𝜒𝜏)    &   (𝜃𝜏)       (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)
 
Theorem4cases 1032 Inference eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 25-Oct-2003.)
((𝜑𝜓) → 𝜒)    &   ((𝜑 ∧ ¬ 𝜓) → 𝜒)    &   ((¬ 𝜑𝜓) → 𝜒)    &   ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)       𝜒
 
Theorem4casesdan 1033 Deduction eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 19-Mar-2013.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)    &   ((𝜑 ∧ (𝜓 ∧ ¬ 𝜒)) → 𝜃)    &   ((𝜑 ∧ (¬ 𝜓𝜒)) → 𝜃)    &   ((𝜑 ∧ (¬ 𝜓 ∧ ¬ 𝜒)) → 𝜃)       (𝜑𝜃)
 
Theoremcases 1034 Case disjunction according to the value of 𝜑. (Contributed by NM, 25-Apr-2019.)
(𝜑 → (𝜓𝜒))    &   𝜑 → (𝜓𝜃))       (𝜓 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))
 
Theoremdedlem0a 1035 Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
(𝜑 → (𝜓 ↔ ((𝜒𝜑) → (𝜓𝜑))))
 
Theoremdedlem0b 1036 Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.)
𝜑 → (𝜓 ↔ ((𝜓𝜑) → (𝜒𝜑))))
 
Theoremdedlema 1037 Lemma for weak deduction theorem. See also ifptru 1065. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
 
Theoremdedlemb 1038 Lemma for weak deduction theorem. See also ifpfal 1066. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
𝜑 → (𝜒 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
 
Theoremcases2 1039 Case disjunction according to the value of 𝜑. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by Wolf Lammen, 28-Feb-2022.)
(((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
 
Theoremcases2ALT 1040 Alternate proof of cases2 1039, not using dedlema 1037 or dedlemb 1038. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by Wolf Lammen, 2-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
 
Theoremdfbi3 1041 An alternate definition of the biconditional. Theorem *5.23 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) (Proof shortened by NM, 29-Oct-2021.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
 
Theorempm5.24 1042 Theorem *5.24 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)
(¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
 
Theorem4exmid 1043 The disjunction of the four possible combinations of two wffs and their negations is always true. A four-way excluded middle (see exmid 888). (Contributed by David Abernethy, 28-Jan-2014.) (Proof shortened by NM, 29-Oct-2021.)
(((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
 
Theoremconsensus 1044 The consensus theorem. This theorem and its dual (with and interchanged) are commonly used in computer logic design to eliminate redundant terms from Boolean expressions. Specifically, we prove that the term (𝜓𝜒) on the left-hand side is redundant. (Contributed by NM, 16-May-2003.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 20-Jan-2013.)
((((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
 
Theorempm4.42 1045 Theorem *4.42 of [WhiteheadRussell] p. 119. See also ifpid 1067. (Contributed by Roy F. Longton, 21-Jun-2005.)
(𝜑 ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))
 
Theoremprlem1 1046 A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
(𝜑 → (𝜂𝜒))    &   (𝜓 → ¬ 𝜃)       (𝜑 → (𝜓 → (((𝜓𝜒) ∨ (𝜃𝜏)) → 𝜂)))
 
Theoremprlem2 1047 A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))
 
Theoremoplem1 1048 A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜓𝜃)    &   (𝜒 → (𝜃𝜏))       (𝜑𝜓)
 
Theoremdn1 1049 A single axiom for Boolean algebra known as DN1. See McCune, Veroff, Fitelson, Harris, Feist, Wos, Short single axioms for Boolean algebra, Journal of Automated Reasoning, 29(1):1--16, 2002. (https://www.cs.unm.edu/~mccune/papers/basax/v12.pdf). (Contributed by Jeff Hankins, 3-Jul-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)
(¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))) ↔ 𝜒)
 
Theorembianir 1050 A closed form of mpbir 232, analogous to pm2.27 42 (assertion). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Roger Witte, 17-Aug-2020.)
((𝜑 ∧ (𝜓𝜑)) → 𝜓)
 
Theoremjaoi2 1051 Inference removing a negated conjunct in a disjunction of an antecedent if this conjunct is part of the disjunction. (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Proof shortened by Wolf Lammen, 21-Sep-2018.)
((𝜑 ∨ (¬ 𝜑𝜒)) → 𝜓)       ((𝜑𝜒) → 𝜓)
 
Theoremjaoi3 1052 Inference separating a disjunct of an antecedent. (Contributed by Alexander van der Vekens, 25-May-2018.)
(𝜑𝜓)    &   ((¬ 𝜑𝜒) → 𝜓)       ((𝜑𝜒) → 𝜓)
 
Theoremornld 1053 Selecting one statement from a disjunction if one of the disjuncted statements is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
(𝜑 → (((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
 
1.2.9  The conditional operator for propositions

This subsection introduces the conditional operator for propositions, denoted by if-(𝜑, 𝜓, 𝜒) (see df-ifp 1055). It is the analogue for propositions of the conditional operator for classes, denoted by if(𝜑, 𝐴, 𝐵) (see df-if 4466).

 
Syntaxwif 1054 Extend class notation to include the conditional operator for propositions.
wff if-(𝜑, 𝜓, 𝜒)
 
Definitiondf-ifp 1055 Definition of the conditional operator for propositions. The expression if-(𝜑, 𝜓, 𝜒) is read "if 𝜑 then 𝜓 else 𝜒". See dfifp2 1056, dfifp3 1057, dfifp4 1058, dfifp5 1059, dfifp6 1060 and dfifp7 1061 for alternate definitions.

This definition (in the form of dfifp2 1056) appears in Section II.24 of [Church] p. 129 (Definition D12 page 132), where it is called "conditioned disjunction". Church's [𝜓, 𝜑, 𝜒] corresponds to our if-(𝜑, 𝜓, 𝜒) (note the permutation of the first two variables).

This form was chosen as the definition rather than dfifp2 1056 for compatibility with intuitionistic logic development: with this form, it is clear that if-(𝜑, 𝜓, 𝜒) implies decidability of 𝜑, which is most often what is wanted.

Church uses the conditional operator as an intermediate step to prove completeness of some systems of connectives. The first result is that the system {if-, ⊤, ⊥} is complete: for the induction step, consider a formula of n+1 variables; single out one variable, say 𝜑; when one sets 𝜑 to True (resp. False), then what remains is a formula of n variables, so by the induction hypothesis it is equivalent to a formula using only the connectives if-, ⊤, ⊥, say 𝜓 (resp. 𝜒); therefore, the formula if-(𝜑, 𝜓, 𝜒) is equivalent to the initial formula of n+1 variables. Now, since { → , ¬ } and similar systems suffice to express the connectives if-, ⊤, ⊥, they are also complete.

(Contributed by BJ, 22-Jun-2019.)

(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
 
Theoremdfifp2 1056 Alternate definition of the conditional operator for propositions. The value of if-(𝜑, 𝜓, 𝜒) is "if 𝜑 then 𝜓, and if not 𝜑 then 𝜒". This version of the definition uses only primitive symbols (→ , ¬ , ∀). This is the definition used in Section II.24 of [Church] p. 129 (Definition D12 page 132) (see comment of df-ifp 1055). (Contributed by BJ, 22-Jun-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
 
Theoremdfifp3 1057 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
 
Theoremdfifp4 1058 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))
 
Theoremdfifp5 1059 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))
 
Theoremdfifp6 1060 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ ¬ (𝜒𝜑)))
 
Theoremdfifp7 1061 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒𝜑) → (𝜑𝜓)))
 
Theoremanifp 1062 The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1063. (Contributed by BJ, 30-Sep-2019.)
((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))
 
Theoremifpor 1063 The conditional operator implies the disjunction of its possible outputs. Dual statement of anifp 1062. (Contributed by BJ, 1-Oct-2019.)
(if-(𝜑, 𝜓, 𝜒) → (𝜓𝜒))
 
Theoremifpn 1064 Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
 
Theoremifptru 1065 Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4471. This is essentially dedlema 1037. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
(𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))
 
Theoremifpfal 1066 Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4474. This is essentially dedlemb 1038. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 25-Jun-2020.)
𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))
 
Theoremifpid 1067 Value of the conditional operator for propositions when the same proposition is returned in either case. Analogue for propositions of ifid 4504. This is essentially pm4.42 1045. (Contributed by BJ, 20-Sep-2019.)
(if-(𝜑, 𝜓, 𝜓) ↔ 𝜓)
 
Theoremcasesifp 1068 Version of cases 1034 expressed using if-. Case disjunction according to the value of 𝜑. One can see this as a proof that the two hypotheses characterize the conditional operator for propositions. For the converses, see ifptru 1065 and ifpfal 1066. (Contributed by BJ, 20-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   𝜑 → (𝜓𝜃))       (𝜓 ↔ if-(𝜑, 𝜒, 𝜃))
 
Theoremifpbi123d 1069 Equality deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.)
(𝜑 → (𝜓𝜏))    &   (𝜑 → (𝜒𝜂))    &   (𝜑 → (𝜃𝜁))       (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))
 
Theoremifpimpda 1070 Separation of the values of the conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 27-Feb-2021.)
((𝜑𝜓) → 𝜒)    &   ((𝜑 ∧ ¬ 𝜓) → 𝜃)       (𝜑 → if-(𝜓, 𝜒, 𝜃))
 
Theorem1fpid3 1071 The value of the conditional operator for propositions is its third argument if the first and second argument imply the third argument. (Contributed by AV, 4-Apr-2021.)
((𝜑𝜓) → 𝜒)       (if-(𝜑, 𝜓, 𝜒) → 𝜒)
 
1.2.10  The weak deduction theorem for propositional calculus

This subsection contains a few results related to the weak deduction theorem in propositional calculus. For the weak deduction theorem in set theory, see the section beginning with dedth 4521. For more information on the weak deduction theorem, see the Weak Deduction Theorem page mmdeduction.html 4521.

 
Theoremelimh 1072 Hypothesis builder for the weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html. (Contributed by NM, 26-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) Commute consequent. (Revised by Steven Nguyen, 27-Apr-2023.)
((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜏𝜒))    &   ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜓) → (𝜏𝜃))    &   𝜃       𝜏
 
TheoremelimhOLD 1073 Obsolete version of elimh 1072 as of 27-Apr-2023. Hypothesis builder for the weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html 1072. (Contributed by NM, 26-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜒𝜏))    &   ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜓) → (𝜃𝜏))    &   𝜃       𝜏
 
Theoremdedt 1074 The weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html. (Contributed by NM, 26-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) Commute consequent. (Revised by Steven Nguyen, 27-Apr-2023.)
((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜏𝜃))    &   𝜏       (𝜒𝜃)
 
TheoremdedtOLD 1075 Obsolete version of dedt 1074 as of 27-Apr-2023. The weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html 1074. (Contributed by NM, 26-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜃𝜏))    &   𝜏       (𝜒𝜃)
 
Theoremcon3ALT 1076 Proof of con3 156 from its associated inference con3i 157 that illustrates the use of the weak deduction theorem dedt 1074. (Contributed by NM, 27-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) Revised dedt 1074 and elimh 1072. (Revised by Steven Nguyen, 27-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
 
Theoremcon3ALTOLD 1077 Obsolete version of con3ALT 1076 as of 27-Apr-2023. Proof of con3 156 from its associated inference con3i 157 that illustrates the use of the weak deduction theorem dedt 1074. (Contributed by NM, 27-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
 
1.2.11  Abbreviated conjunction and disjunction of three wff's
 
Syntaxw3o 1078 Extend wff definition to include three-way disjunction ('or').
wff (𝜑𝜓𝜒)
 
Syntaxw3a 1079 Extend wff definition to include three-way conjunction ('and').
wff (𝜑𝜓𝜒)
 
Definitiondf-3or 1080 Define disjunction ('or') of three wff's. Definition *2.33 of [WhiteheadRussell] p. 105. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law orass 915. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
 
Definitiondf-3an 1081 Define conjunction ('and') of three wff's. Definition *4.34 of [WhiteheadRussell] p. 118. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law anass 469. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
 
Theorem3orass 1082 Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑 ∨ (𝜓𝜒)))
 
Theorem3orel1 1083 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)
𝜑 → ((𝜑𝜓𝜒) → (𝜓𝜒)))
 
Theorem3orrot 1084 Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
((𝜑𝜓𝜒) ↔ (𝜓𝜒𝜑))
 
Theorem3orcoma 1085 Commutation law for triple disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓𝜒) ↔ (𝜓𝜑𝜒))
 
Theorem3orcomb 1086 Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.) (Proof shortened by Wolf Lammen, 8-Apr-2022.)
((𝜑𝜓𝜒) ↔ (𝜑𝜒𝜓))
 
Theorem3anass 1087 Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
 
Theorem3anan12 1088 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised to shorten 3ancoma 1090 by Wolf Lammen, 5-Jun-2022.)
((𝜑𝜓𝜒) ↔ (𝜓 ∧ (𝜑𝜒)))
 
Theorem3anan32 1089 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))
 
Theorem3ancoma 1090 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 5-Jun-2022.)
((𝜑𝜓𝜒) ↔ (𝜓𝜑𝜒))
 
Theorem3ancomb 1091 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.) (Revised to shorten 3anrot 1092 by Wolf Lammen, 9-Jun-2022.)
((𝜑𝜓𝜒) ↔ (𝜑𝜒𝜓))
 
Theorem3anrot 1092 Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.) (Proof shortened by Wolf Lammen, 9-Jun-2022.)
((𝜑𝜓𝜒) ↔ (𝜓𝜒𝜑))
 
Theorem3anrev 1093 Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜒𝜓𝜑))
 
Theoremanandi3 1094 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
 
Theoremanandi3r 1095 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))
 
Theorem3anidm 1096 Idempotent law for conjunction. (Contributed by Peter Mazsa, 17-Oct-2023.)
((𝜑𝜑𝜑) ↔ 𝜑)
 
Theorem3an4anass 1097 Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
(((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
 
Theorem3ioran 1098 Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
(¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
 
Theorem3ianor 1099 Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Revised by Wolf Lammen, 8-Apr-2022.)
(¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
 
Theorem3anor 1100 Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Wolf Lammen, 8-Apr-2022.)
((𝜑𝜓𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
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