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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | pm5.53 1001 | Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
⊢ ((((𝜑 ∨ 𝜓) ∨ 𝜒) → 𝜃) ↔ (((𝜑 → 𝜃) ∧ (𝜓 → 𝜃)) ∧ (𝜒 → 𝜃))) | ||
Theorem | ordi 1002 | 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 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) | ||
Theorem | ordir 1003 | Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒))) | ||
Theorem | andi 1004 | 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 ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒))) | ||
Theorem | andir 1005 | Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))) | ||
Theorem | orddi 1006 | Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.) |
⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)) ↔ (((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜃)) ∧ ((𝜓 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)))) | ||
Theorem | anddi 1007 | Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.) |
⊢ (((𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜃)) ↔ (((𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)) ∨ ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)))) | ||
Theorem | pm5.17 1008 | Theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Jan-2013.) |
⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)) | ||
Theorem | pm5.15 1009 | Theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 15-Oct-2013.) |
⊢ ((𝜑 ↔ 𝜓) ∨ (𝜑 ↔ ¬ 𝜓)) | ||
Theorem | pm5.16 1010 | Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 17-Oct-2013.) |
⊢ ¬ ((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓)) | ||
Theorem | xor 1011 | Two ways to express exclusive disjunction (df-xor 1502). Theorem *5.22 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 22-Jan-2013.) |
⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))) | ||
Theorem | nbi2 1012 | Two ways to express "exclusive or". (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Jan-2013.) |
⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) | ||
Theorem | xordi 1013 | Conjunction distributes over exclusive-or, using ¬ (𝜑 ↔ 𝜓) 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 1518 does hold in it. (Contributed by NM, 3-Oct-2008.) |
⊢ ((𝜑 ∧ ¬ (𝜓 ↔ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) | ||
Theorem | pm5.54 1014 | Theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 7-Nov-2013.) |
⊢ (((𝜑 ∧ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∧ 𝜓) ↔ 𝜓)) | ||
Theorem | pm5.62 1015 | Theorem *5.62 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 21-Jun-2005.) |
⊢ (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)) | ||
Theorem | pm5.63 1016 | Theorem *5.63 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 25-Dec-2012.) |
⊢ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ (¬ 𝜑 ∧ 𝜓))) | ||
Theorem | niabn 1017 | Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.) |
⊢ 𝜑 ⇒ ⊢ (¬ 𝜓 → ((𝜒 ∧ 𝜓) ↔ ¬ 𝜑)) | ||
Theorem | ninba 1018 | Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.) |
⊢ 𝜑 ⇒ ⊢ (¬ 𝜓 → (¬ 𝜑 ↔ (𝜒 ∧ 𝜓))) | ||
Theorem | pm4.43 1019 | Theorem *4.43 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 26-Nov-2012.) |
⊢ (𝜑 ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ ¬ 𝜓))) | ||
Theorem | pm4.82 1020 | Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑) | ||
Theorem | pm4.83 1021 | Theorem *4.83 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)) ↔ 𝜓) | ||
Theorem | pclem6 1022 | Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Nov-2012.) |
⊢ ((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓) | ||
Theorem | bigolden 1023 | Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.) |
⊢ (((𝜑 ∧ 𝜓) ↔ 𝜑) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓))) | ||
Theorem | pm5.71 1024 | Theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.) |
⊢ ((𝜓 → ¬ 𝜒) → (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ 𝜒))) | ||
Theorem | pm5.75 1025 | 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 ↔ (𝜓 ∨ 𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒)) | ||
Theorem | ecase2d 1026 | Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Dec-2012.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒)) & ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜃)) & ⊢ (𝜑 → (𝜏 ∨ (𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → 𝜏) | ||
Theorem | ecase3 1027 | Inference for elimination by cases. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 26-Nov-2012.) |
⊢ (𝜑 → 𝜒) & ⊢ (𝜓 → 𝜒) & ⊢ (¬ (𝜑 ∨ 𝜓) → 𝜒) ⇒ ⊢ 𝜒 | ||
Theorem | ecase 1028 | Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.) |
⊢ (¬ 𝜑 → 𝜒) & ⊢ (¬ 𝜓 → 𝜒) & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ 𝜒 | ||
Theorem | ecase3d 1029 | Deduction for elimination by cases. (Contributed by NM, 2-May-1996.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜑 → (𝜒 → 𝜃)) & ⊢ (𝜑 → (¬ (𝜓 ∨ 𝜒) → 𝜃)) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | ecased 1030 | Deduction for elimination by cases. (Contributed by NM, 8-Oct-2012.) |
⊢ (𝜑 → (¬ 𝜓 → 𝜃)) & ⊢ (𝜑 → (¬ 𝜒 → 𝜃)) & ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | ecase3ad 1031 | Deduction for elimination by cases. (Contributed by NM, 24-May-2013.) |
⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜑 → (𝜒 → 𝜃)) & ⊢ (𝜑 → ((¬ 𝜓 ∧ ¬ 𝜒) → 𝜃)) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | ccase 1032 | Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜏) & ⊢ ((𝜒 ∧ 𝜓) → 𝜏) & ⊢ ((𝜑 ∧ 𝜃) → 𝜏) & ⊢ ((𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏) | ||
Theorem | ccased 1033 | Deduction for combining cases. (Contributed by NM, 9-May-2004.) |
⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜂)) & ⊢ (𝜑 → ((𝜃 ∧ 𝜒) → 𝜂)) & ⊢ (𝜑 → ((𝜓 ∧ 𝜏) → 𝜂)) & ⊢ (𝜑 → ((𝜃 ∧ 𝜏) → 𝜂)) ⇒ ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∧ (𝜒 ∨ 𝜏)) → 𝜂)) | ||
Theorem | ccase2 1034 | Inference for combining cases. (Contributed by NM, 29-Jul-1999.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜏) & ⊢ (𝜒 → 𝜏) & ⊢ (𝜃 → 𝜏) ⇒ ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏) | ||
Theorem | 4cases 1035 | Inference eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 25-Oct-2003.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒) & ⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒) ⇒ ⊢ 𝜒 | ||
Theorem | 4casesdan 1036 | Deduction eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 19-Mar-2013.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) & ⊢ ((𝜑 ∧ (𝜓 ∧ ¬ 𝜒)) → 𝜃) & ⊢ ((𝜑 ∧ (¬ 𝜓 ∧ 𝜒)) → 𝜃) & ⊢ ((𝜑 ∧ (¬ 𝜓 ∧ ¬ 𝜒)) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | cases 1037 | Case disjunction according to the value of 𝜑. (Contributed by NM, 25-Apr-2019.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (¬ 𝜑 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃))) | ||
Theorem | dedlem0a 1038 | 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 ↔ ((𝜒 → 𝜑) → (𝜓 ∧ 𝜑)))) | ||
Theorem | dedlem0b 1039 | Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.) |
⊢ (¬ 𝜑 → (𝜓 ↔ ((𝜓 → 𝜑) → (𝜒 ∧ 𝜑)))) | ||
Theorem | dedlema 1040 | Lemma for weak deduction theorem. See also ifptru 1068. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ (𝜑 → (𝜓 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) | ||
Theorem | dedlemb 1041 | Lemma for weak deduction theorem. See also ifpfal 1069. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) | ||
Theorem | cases2 1042 | Case disjunction according to the value of 𝜑. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by Wolf Lammen, 28-Feb-2022.) |
⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) | ||
Theorem | cases2ALT 1043 | Alternate proof of cases2 1042, not using dedlema 1040 or dedlemb 1041. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by Wolf Lammen, 2-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) | ||
Theorem | dfbi3 1044 | 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.) |
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))) | ||
Theorem | pm5.24 1045 | Theorem *5.24 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))) | ||
Theorem | 4exmid 1046 | The disjunction of the four possible combinations of two wffs and their negations is always true. A four-way excluded middle (see exmid 891). (Contributed by David Abernethy, 28-Jan-2014.) (Proof shortened by NM, 29-Oct-2021.) |
⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))) | ||
Theorem | consensus 1047 | 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 (𝜓 ∧ 𝜒) 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.) |
⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) | ||
Theorem | pm4.42 1048 | Theorem *4.42 of [WhiteheadRussell] p. 119. See also ifpid 1070. (Contributed by Roy F. Longton, 21-Jun-2005.) |
⊢ (𝜑 ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓))) | ||
Theorem | prlem1 1049 | 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 → (𝜂 ↔ 𝜒)) & ⊢ (𝜓 → ¬ 𝜃) ⇒ ⊢ (𝜑 → (𝜓 → (((𝜓 ∧ 𝜒) ∨ (𝜃 ∧ 𝜏)) → 𝜂))) | ||
Theorem | prlem2 1050 | 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.) |
⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∧ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)))) | ||
Theorem | oplem1 1051 | A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.) |
⊢ (𝜑 → (𝜓 ∨ 𝜒)) & ⊢ (𝜑 → (𝜃 ∨ 𝜏)) & ⊢ (𝜓 ↔ 𝜃) & ⊢ (𝜒 → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | dn1 1052 | 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.) |
⊢ (¬ (¬ (¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))) ↔ 𝜒) | ||
Theorem | bianir 1053 | A closed form of mpbir 233, analogous to pm2.27 42 (assertion). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Roger Witte, 17-Aug-2020.) |
⊢ ((𝜑 ∧ (𝜓 ↔ 𝜑)) → 𝜓) | ||
Theorem | jaoi2 1054 | 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 ∨ (¬ 𝜑 ∧ 𝜒)) → 𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜒) → 𝜓) | ||
Theorem | jaoi3 1055 | Inference separating a disjunct of an antecedent. (Contributed by Alexander van der Vekens, 25-May-2018.) |
⊢ (𝜑 → 𝜓) & ⊢ ((¬ 𝜑 ∧ 𝜒) → 𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜒) → 𝜓) | ||
Theorem | ornld 1056 | 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 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏)) | ||
This subsection introduces the conditional operator for propositions, denoted by if-(𝜑, 𝜓, 𝜒) (see df-ifp 1058). It is the analogue for propositions of the conditional operator for classes, denoted by if(𝜑, 𝐴, 𝐵) (see df-if 4471). | ||
Syntax | wif 1057 | Extend class notation to include the conditional operator for propositions. |
wff if-(𝜑, 𝜓, 𝜒) | ||
Definition | df-ifp 1058 |
Definition of the conditional operator for propositions. The expression
if-(𝜑,
𝜓, 𝜒) is read "if 𝜑 then
𝜓
else 𝜒".
See dfifp2 1059, dfifp3 1060, dfifp4 1061, dfifp5 1062, dfifp6 1063 and dfifp7 1064 for
alternate definitions.
This definition (in the form of dfifp2 1059) 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 1059 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) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) | ||
Theorem | dfifp2 1059 | 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 1058). (Contributed by BJ, 22-Jun-2019.) |
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) | ||
Theorem | dfifp3 1060 | Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.) |
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))) | ||
Theorem | dfifp4 1061 | Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.) |
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) | ||
Theorem | dfifp5 1062 | Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.) |
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 → 𝜒))) | ||
Theorem | dfifp6 1063 | Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.) |
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜒 → 𝜑))) | ||
Theorem | dfifp7 1064 | Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.) |
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒 → 𝜑) → (𝜑 ∧ 𝜓))) | ||
Theorem | anifp 1065 | The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1066. (Contributed by BJ, 30-Sep-2019.) |
⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒)) | ||
Theorem | ifpor 1066 | The conditional operator implies the disjunction of its possible outputs. Dual statement of anifp 1065. (Contributed by BJ, 1-Oct-2019.) |
⊢ (if-(𝜑, 𝜓, 𝜒) → (𝜓 ∨ 𝜒)) | ||
Theorem | ifpn 1067 | Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.) |
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓)) | ||
Theorem | ifptru 1068 | Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4476. This is essentially dedlema 1040. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 10-Jul-2020.) |
⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓)) | ||
Theorem | ifpfal 1069 | Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4479. This is essentially dedlemb 1041. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 25-Jun-2020.) |
⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒)) | ||
Theorem | ifpid 1070 | Value of the conditional operator for propositions when the same proposition is returned in either case. Analogue for propositions of ifid 4509. This is essentially pm4.42 1048. (Contributed by BJ, 20-Sep-2019.) |
⊢ (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓) | ||
Theorem | casesifp 1071 | Version of cases 1037 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 1068 and ifpfal 1069. (Contributed by BJ, 20-Sep-2019.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (¬ 𝜑 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝜓 ↔ if-(𝜑, 𝜒, 𝜃)) | ||
Theorem | ifpbi123d 1072 | Equality deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 17-Apr-2024.) |
⊢ (𝜑 → (𝜓 ↔ 𝜏)) & ⊢ (𝜑 → (𝜒 ↔ 𝜂)) & ⊢ (𝜑 → (𝜃 ↔ 𝜁)) ⇒ ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁))) | ||
Theorem | ifpbi123dOLD 1073 | Obsolete version of ifpbi123d 1072 as of 17-Apr-2024. (Contributed by AV, 30-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 ↔ 𝜏)) & ⊢ (𝜑 → (𝜒 ↔ 𝜂)) & ⊢ (𝜑 → (𝜃 ↔ 𝜁)) ⇒ ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁))) | ||
Theorem | ifpimpda 1074 | Separation of the values of the conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 27-Feb-2021.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜃) ⇒ ⊢ (𝜑 → if-(𝜓, 𝜒, 𝜃)) | ||
Theorem | 1fpid3 1075 | 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.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ (if-(𝜑, 𝜓, 𝜒) → 𝜒) | ||
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 4526. For more information on the weak deduction theorem, see the Weak Deduction Theorem page mmdeduction.html 4526. | ||
Theorem | elimh 1076 | 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) → (𝜏 ↔ 𝜒)) & ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜓) → (𝜏 ↔ 𝜃)) & ⊢ 𝜃 ⇒ ⊢ 𝜏 | ||
Theorem | elimhOLD 1077 | Obsolete version of elimh 1076 as of 27-Apr-2023. Hypothesis builder for the weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html 1076. (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) → (𝜒 ↔ 𝜏)) & ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜓) → (𝜃 ↔ 𝜏)) & ⊢ 𝜃 ⇒ ⊢ 𝜏 | ||
Theorem | dedt 1078 | 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) → (𝜏 ↔ 𝜃)) & ⊢ 𝜏 ⇒ ⊢ (𝜒 → 𝜃) | ||
Theorem | dedtOLD 1079 | Obsolete version of dedt 1078 as of 27-Apr-2023. The weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html 1078. (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) → (𝜃 ↔ 𝜏)) & ⊢ 𝜏 ⇒ ⊢ (𝜒 → 𝜃) | ||
Theorem | con3ALT 1080 | Proof of con3 156 from its associated inference con3i 157 that illustrates the use of the weak deduction theorem dedt 1078. (Contributed by NM, 27-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) Revised dedt 1078 and elimh 1076. (Revised by Steven Nguyen, 27-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | ||
Theorem | con3ALTOLD 1081 | Obsolete version of con3ALT 1080 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 1078. (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.) |
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | ||
Syntax | w3o 1082 | Extend wff definition to include three-way disjunction ('or'). |
wff (𝜑 ∨ 𝜓 ∨ 𝜒) | ||
Syntax | w3a 1083 | Extend wff definition to include three-way conjunction ('and'). |
wff (𝜑 ∧ 𝜓 ∧ 𝜒) | ||
Definition | df-3or 1084 | 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 918. (Contributed by NM, 8-Apr-1994.) |
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒)) | ||
Definition | df-3an 1085 | 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 471. (Contributed by NM, 8-Apr-1994.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | ||
Theorem | 3orass 1086 | Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.) |
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒))) | ||
Theorem | 3orel1 1087 | Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.) |
⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓 ∨ 𝜒) → (𝜓 ∨ 𝜒))) | ||
Theorem | 3orrot 1088 | Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.) |
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒 ∨ 𝜑)) | ||
Theorem | 3orcoma 1089 | Commutation law for triple disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜓 ∨ 𝜑 ∨ 𝜒)) | ||
Theorem | 3orcomb 1090 | Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.) (Proof shortened by Wolf Lammen, 8-Apr-2022.) |
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜑 ∨ 𝜒 ∨ 𝜓)) | ||
Theorem | 3anass 1091 | Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒))) | ||
Theorem | 3anan12 1092 | 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 1094 by Wolf Lammen, 5-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒))) | ||
Theorem | 3anan32 1093 | Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ 𝜓)) | ||
Theorem | 3ancoma 1094 | Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 5-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒)) | ||
Theorem | 3ancomb 1095 | Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.) (Revised to shorten 3anrot 1096 by Wolf Lammen, 9-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓)) | ||
Theorem | 3anrot 1096 | Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.) (Proof shortened by Wolf Lammen, 9-Jun-2022.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒 ∧ 𝜑)) | ||
Theorem | 3anrev 1097 | Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓 ∧ 𝜑)) | ||
Theorem | anandi3 1098 | Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒))) | ||
Theorem | anandi3r 1099 | Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓))) | ||
Theorem | 3anidm 1100 | Idempotent law for conjunction. (Contributed by Peter Mazsa, 17-Oct-2023.) |
⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) ↔ 𝜑) |
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