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Theorem List for Metamath Proof Explorer - 1001-1100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempm4.79 1001 Theorem *4.79 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2013.)
(((𝜓𝜑) ∨ (𝜒𝜑)) ↔ ((𝜓𝜒) → 𝜑))

Theorempm5.53 1002 Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((((𝜑𝜓) ∨ 𝜒) → 𝜃) ↔ (((𝜑𝜃) ∧ (𝜓𝜃)) ∧ (𝜒𝜃)))

Theoremordi 1003 Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 28-Nov-2013.)
((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Theoremordir 1004 Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))

Theoremandi 1005 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 1006 Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))

Theoremorddi 1007 Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ (((𝜑𝜒) ∧ (𝜑𝜃)) ∧ ((𝜓𝜒) ∧ (𝜓𝜃))))

Theoremanddi 1008 Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ (((𝜑𝜒) ∨ (𝜑𝜃)) ∨ ((𝜓𝜒) ∨ (𝜓𝜃))))

Theorempm5.17 1009 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 1010 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 1011 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 1012 Two ways to express exclusive disjunction (df-xor 1503). 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 1013 Two ways to express "exclusive or". (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)
(¬ (𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))

Theoremxordi 1014 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 1519 does hold in it. (Contributed by NM, 3-Oct-2008.)
((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))

Theorempm5.54 1015 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 1016 Theorem *5.62 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 21-Jun-2005.)
(((𝜑𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))

Theorempm5.63 1017 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 1018 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
𝜑       𝜓 → ((𝜒𝜓) ↔ ¬ 𝜑))

Theoremninba 1019 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
𝜑       𝜓 → (¬ 𝜑 ↔ (𝜒𝜓)))

Theorempm4.43 1020 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 1021 Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)

Theorempm4.83 1022 Theorem *4.83 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (¬ 𝜑𝜓)) ↔ 𝜓)

Theorempclem6 1023 Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Nov-2012.)
((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)

Theorembigolden 1024 Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)
(((𝜑𝜓) ↔ 𝜑) ↔ (𝜓 ↔ (𝜑𝜓)))

Theorempm5.71 1025 Theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.)
((𝜓 → ¬ 𝜒) → (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑𝜒)))

Theorempm5.75 1026 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 1027 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 1028 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 1029 Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)
𝜑𝜒)    &   𝜓𝜒)    &   ((𝜑𝜓) → 𝜒)       𝜒

Theoremecase3d 1030 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 1031 Deduction for elimination by cases. (Contributed by NM, 8-Oct-2012.)
(𝜑 → (¬ 𝜓𝜃))    &   (𝜑 → (¬ 𝜒𝜃))    &   (𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑𝜃)

Theoremecase3ad 1032 Deduction for elimination by cases. (Contributed by NM, 24-May-2013.)
(𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → ((¬ 𝜓 ∧ ¬ 𝜒) → 𝜃))       (𝜑𝜃)

Theoremccase 1033 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 1034 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 1035 Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
((𝜑𝜓) → 𝜏)    &   (𝜒𝜏)    &   (𝜃𝜏)       (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)

Theorem4cases 1036 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 1037 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 1038 Case disjunction according to the value of 𝜑. (Contributed by NM, 25-Apr-2019.)
(𝜑 → (𝜓𝜒))    &   𝜑 → (𝜓𝜃))       (𝜓 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))

Theoremdedlem0a 1039 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 1040 Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.)
𝜑 → (𝜓 ↔ ((𝜓𝜑) → (𝜒𝜑))))

Theoremdedlema 1041 Lemma for weak deduction theorem. See also ifptru 1071. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))

Theoremdedlemb 1042 Lemma for weak deduction theorem. See also ifpfal 1072. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
𝜑 → (𝜒 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))

Theoremcases2 1043 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 1044 Alternate proof of cases2 1043, not using dedlema 1041 or dedlemb 1042. (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 1045 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 1046 Theorem *5.24 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)
(¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))

Theorem4exmid 1047 The disjunction of the four possible combinations of two wffs and their negations is always true. A four-way excluded middle (see exmid 892). (Contributed by David Abernethy, 28-Jan-2014.) (Proof shortened by NM, 29-Oct-2021.)
(((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))

Theoremconsensus 1048 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 1049 Theorem *4.42 of [WhiteheadRussell] p. 119. See also ifpid 1073. (Contributed by Roy F. Longton, 21-Jun-2005.)
(𝜑 ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))

Theoremprlem1 1050 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 1051 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 1052 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 1053 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 1054 A closed form of mpbir 234, 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 1055 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 1056 Inference separating a disjunct of an antecedent. (Contributed by Alexander van der Vekens, 25-May-2018.)
(𝜑𝜓)    &   ((¬ 𝜑𝜒) → 𝜓)       ((𝜑𝜒) → 𝜓)

Theoremornld 1057 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 1059). It is the analogue for propositions of the conditional operator for classes, denoted by if(𝜑, 𝐴, 𝐵) (see df-if 4426).

Syntaxwif 1058 Extend wff notation to include the conditional operator for propositions.
wff if-(𝜑, 𝜓, 𝜒)

Definitiondf-ifp 1059 Definition of the conditional operator for propositions. The expression if-(𝜑, 𝜓, 𝜒) is read "if 𝜑 then 𝜓 else 𝜒". See dfifp2 1060, dfifp3 1061, dfifp4 1062, dfifp5 1063, dfifp6 1064 and dfifp7 1065 for alternate definitions.

This definition (in the form of dfifp2 1060) 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 1060 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 1060 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 is the definition used in Section II.24 of [Church] p. 129 (Definition D12 page 132) (see comment of df-ifp 1059). (Contributed by BJ, 22-Jun-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))

Theoremdfifp3 1061 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Theoremdfifp4 1062 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))

Theoremdfifp5 1063 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))

Theoremdfifp6 1064 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ ¬ (𝜒𝜑)))

Theoremdfifp7 1065 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒𝜑) → (𝜑𝜓)))

Theoremifpdfbi 1066 Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020.) (Proof shortened by Wolf Lammen, 30-Apr-2024.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))

Theoremanifp 1067 The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1068. (Contributed by BJ, 30-Sep-2019.)
((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))

Theoremifpor 1068 The conditional operator implies the disjunction of its possible outputs. Dual statement of anifp 1067. (Contributed by BJ, 1-Oct-2019.)
(if-(𝜑, 𝜓, 𝜒) → (𝜓𝜒))

Theoremifpn 1069 Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.) (Proof shortened by Wolf Lammen, 5-May-2024.)
(if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))

TheoremifpnOLD 1070 Obsolete version of ifpn 1069 as of 5-May-2024.. (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))

Theoremifptru 1071 Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4431. This is essentially dedlema 1041. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
(𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))

Theoremifpfal 1072 Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4434. This is essentially dedlemb 1042. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 25-Jun-2020.)
𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))

Theoremifpid 1073 Value of the conditional operator for propositions when the same proposition is returned in either case. Analogue for propositions of ifid 4464. This is essentially pm4.42 1049. (Contributed by BJ, 20-Sep-2019.)
(if-(𝜑, 𝜓, 𝜓) ↔ 𝜓)

Theoremcasesifp 1074 Version of cases 1038 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 1071 and ifpfal 1072. (Contributed by BJ, 20-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   𝜑 → (𝜓𝜃))       (𝜓 ↔ if-(𝜑, 𝜒, 𝜃))

Theoremifpbi123d 1075 Equivalence deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 17-Apr-2024.)
(𝜑 → (𝜓𝜏))    &   (𝜑 → (𝜒𝜂))    &   (𝜑 → (𝜃𝜁))       (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Theoremifpbi123dOLD 1076 Obsolete version of ifpbi123d 1075 as of 17-Apr-2024. (Contributed by AV, 30-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜏))    &   (𝜑 → (𝜒𝜂))    &   (𝜑 → (𝜃𝜁))       (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Theoremifpbi23d 1077 Equivalence deduction for conditional operator for propositions. Convenience theorem for a frequent case. (Contributed by Wolf Lammen, 28-Apr-2024.)
(𝜑 → (𝜒𝜂))    &   (𝜑 → (𝜃𝜁))       (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜓, 𝜂, 𝜁)))

Theoremifpimpda 1078 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 1079 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 4481. For more information on the weak deduction theorem, see the Weak Deduction Theorem page mmdeduction.html 4481.

Theoremelimh 1080 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

Theoremdedt 1081 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)

Theoremcon3ALT 1082 Proof of con3 156 from its associated inference con3i 157 that illustrates the use of the weak deduction theorem dedt 1081. (Contributed by NM, 27-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) Revised dedt 1081 and elimh 1080. (Revised by Steven Nguyen, 27-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

1.2.11  Abbreviated conjunction and disjunction of three wff's

Syntaxw3o 1083 Extend wff definition to include three-way disjunction ('or').
wff (𝜑𝜓𝜒)

Syntaxw3a 1084 Extend wff definition to include three-way conjunction ('and').
wff (𝜑𝜓𝜒)

Definitiondf-3or 1085 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 919. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))

Definitiondf-3an 1086 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 472. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))

Theorem3orass 1087 Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑 ∨ (𝜓𝜒)))

Theorem3orel1 1088 Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)
𝜑 → ((𝜑𝜓𝜒) → (𝜓𝜒)))

Theorem3orrot 1089 Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
((𝜑𝜓𝜒) ↔ (𝜓𝜒𝜑))

Theorem3orcoma 1090 Commutation law for triple disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓𝜒) ↔ (𝜓𝜑𝜒))

Theorem3orcomb 1091 Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.) (Proof shortened by Wolf Lammen, 8-Apr-2022.)
((𝜑𝜓𝜒) ↔ (𝜑𝜒𝜓))

Theorem3anass 1092 Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))

Theorem3anan12 1093 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 1095 by Wolf Lammen, 5-Jun-2022.)
((𝜑𝜓𝜒) ↔ (𝜓 ∧ (𝜑𝜒)))

Theorem3anan32 1094 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))

Theorem3ancoma 1095 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 5-Jun-2022.)
((𝜑𝜓𝜒) ↔ (𝜓𝜑𝜒))

Theorem3ancomb 1096 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.) (Revised to shorten 3anrot 1097 by Wolf Lammen, 9-Jun-2022.)
((𝜑𝜓𝜒) ↔ (𝜑𝜒𝜓))

Theorem3anrot 1097 Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.) (Proof shortened by Wolf Lammen, 9-Jun-2022.)
((𝜑𝜓𝜒) ↔ (𝜓𝜒𝜑))

Theorem3anrev 1098 Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜒𝜓𝜑))

Theoremanandi3 1099 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Theoremanandi3r 1100 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))

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