P. 11 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	pm4.79 1001	Theorem *4.79 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2013.)
⊢ (((𝜓 → 𝜑) ∨ (𝜒 → 𝜑)) ↔ ((𝜓 ∧ 𝜒) → 𝜑))
Theorem	pm5.53 1002	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 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 ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
Theorem	ordir 1004	Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
⊢ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))
Theorem	andi 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 ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
Theorem	andir 1006	Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))
Theorem	orddi 1007	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 1008	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 1009	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 1010	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 1011	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 1012	Two ways to express exclusive disjunction (df-xor 1507). 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 1013	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 1014	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 1523 does hold in it. (Contributed by NM, 3-Oct-2008.)
⊢ ((𝜑 ∧ ¬ (𝜓 ↔ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))
Theorem	pm5.54 1015	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 1016	Theorem *5.62 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 21-Jun-2005.)
⊢ (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))
Theorem	pm5.63 1017	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 1018	Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜓 → ((𝜒 ∧ 𝜓) ↔ ¬ 𝜑))
Theorem	ninba 1019	Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜓 → (¬ 𝜑 ↔ (𝜒 ∧ 𝜓)))
Theorem	pm4.43 1020	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 1021	Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)
Theorem	pm4.83 1022	Theorem *4.83 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)) ↔ 𝜓)
Theorem	pclem6 1023	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 1024	Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)
⊢ (((𝜑 ∧ 𝜓) ↔ 𝜑) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓)))
Theorem	pm5.71 1025	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 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 ↔ (𝜓 ∨ 𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))
Theorem	ecase2d 1027	Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Sep-2024.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒))    &   ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜃))    &   ⊢ (𝜑 → (𝜏 ∨ (𝜒 ∨ 𝜃)))    ⇒   ⊢ (𝜑 → 𝜏)
Theorem	ecase2dOLD 1028	Obsolete version of ecase2d 1027 as of 19-Sep-2024. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Dec-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒))    &   ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜃))    &   ⊢ (𝜑 → (𝜏 ∨ (𝜒 ∨ 𝜃)))    ⇒   ⊢ (𝜑 → 𝜏)
Theorem	ecase3 1029	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 1030	Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)
⊢ (¬ 𝜑 → 𝜒)    &   ⊢ (¬ 𝜓 → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ 𝜒
Theorem	ecase3d 1031	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 1032	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 1033	Deduction for elimination by cases. (Contributed by NM, 24-May-2013.) (Proof shortened by Wolf Lammen, 20-Sep-2024.)
⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜑 → (𝜒 → 𝜃))    &   ⊢ (𝜑 → ((¬ 𝜓 ∧ ¬ 𝜒) → 𝜃))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	ecase3adOLD 1034	Obsolete version of ecase3ad 1033 as of 20-Sep-2024. (Contributed by NM, 24-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜑 → (𝜒 → 𝜃))    &   ⊢ (𝜑 → ((¬ 𝜓 ∧ ¬ 𝜒) → 𝜃))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	ccase 1035	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 1036	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 1037	Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
⊢ ((𝜑 ∧ 𝜓) → 𝜏)    &   ⊢ (𝜒 → 𝜏)    &   ⊢ (𝜃 → 𝜏)    ⇒   ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏)
Theorem	4cases 1038	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 1039	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 1040	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 1041	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 1042	Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.)
⊢ (¬ 𝜑 → (𝜓 ↔ ((𝜓 → 𝜑) → (𝜒 ∧ 𝜑))))
Theorem	dedlema 1043	Lemma for weak deduction theorem. See also ifptru 1073. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ (𝜑 → (𝜓 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Theorem	dedlemb 1044	Lemma for weak deduction theorem. See also ifpfal 1074. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Theorem	cases2 1045	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 1046	Alternate proof of cases2 1045, not using dedlema 1043 or dedlemb 1044. (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 1047	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 1048	Theorem *5.24 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
Theorem	4exmid 1049	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.)
⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
Theorem	consensus 1050	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 1051	Theorem *4.42 of [WhiteheadRussell] p. 119. See also ifpid 1075. (Contributed by Roy F. Longton, 21-Jun-2005.)
⊢ (𝜑 ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))
Theorem	prlem1 1052	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 1053	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 1054	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 1055	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 1056	A closed form of mpbir 230, 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 1057	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 1058	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 1059	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))
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 1061). It is the analogue for propositions of the conditional operator for classes, denoted by if(𝜑, 𝐴, 𝐵) (see df-if 4460).
Syntax	wif 1060	Extend wff notation to include the conditional operator for propositions.
wff if-(𝜑, 𝜓, 𝜒)
Definition	df-ifp 1061	Definition of the conditional operator for propositions. The expression if-(𝜑, 𝜓, 𝜒) is read "if 𝜑 then 𝜓 else 𝜒". See dfifp2 1062, dfifp3 1063, dfifp4 1064, dfifp5 1065, dfifp6 1066 and dfifp7 1067 for alternate definitions. This definition (in the form of dfifp2 1062) 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 1062 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 1062	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 1061). (Contributed by BJ, 22-Jun-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
Theorem	dfifp3 1063	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
Theorem	dfifp4 1064	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
Theorem	dfifp5 1065	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 → 𝜒)))
Theorem	dfifp6 1066	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜒 → 𝜑)))
Theorem	dfifp7 1067	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒 → 𝜑) → (𝜑 ∧ 𝜓)))
Theorem	ifpdfbi 1068	Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020.) (Proof shortened by Wolf Lammen, 30-Apr-2024.)
⊢ ((𝜑 ↔ 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))
Theorem	anifp 1069	The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1070. (Contributed by BJ, 30-Sep-2019.)
⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
Theorem	ifpor 1070	The conditional operator implies the disjunction of its possible outputs. Dual statement of anifp 1069. (Contributed by BJ, 1-Oct-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) → (𝜓 ∨ 𝜒))
Theorem	ifpn 1071	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))
Theorem	ifpnOLD 1072	Obsolete version of ifpn 1071 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))
Theorem	ifptru 1073	Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4465. This is essentially dedlema 1043. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))
Theorem	ifpfal 1074	Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4468. This is essentially dedlemb 1044. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 25-Jun-2020.)
⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))
Theorem	ifpid 1075	Value of the conditional operator for propositions when the same proposition is returned in either case. Analogue for propositions of ifid 4499. This is essentially pm4.42 1051. (Contributed by BJ, 20-Sep-2019.)
⊢ (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓)
Theorem	casesifp 1076	Version of cases 1040 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 1073 and ifpfal 1074. (Contributed by BJ, 20-Sep-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (¬ 𝜑 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜓 ↔ if-(𝜑, 𝜒, 𝜃))
Theorem	ifpbi123d 1077	Equivalence 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 1078	Obsolete version of ifpbi123d 1077 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	ifpbi23d 1079	Equivalence deduction for conditional operator for propositions. Convenience theorem for a frequent case. (Contributed by Wolf Lammen, 28-Apr-2024.)
⊢ (𝜑 → (𝜒 ↔ 𝜂))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜁))    ⇒   ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜓, 𝜂, 𝜁)))
Theorem	ifpimpda 1080	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 1081	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)
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 4517. For more information on the weak deduction theorem, see the Weak Deduction Theorem page mmdeduction.html 4517.
Theorem	elimh 1082	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	dedt 1083	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	con3ALT 1084	Proof of con3 153 from its associated inference con3i 154 that illustrates the use of the weak deduction theorem dedt 1083. (Contributed by NM, 27-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) Revised dedt 1083 and elimh 1082. (Revised by Steven Nguyen, 27-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
1.2.11  Abbreviated conjunction and disjunction of three wff's
Syntax	w3o 1085	Extend wff definition to include three-way disjunction ('or').
wff (𝜑 ∨ 𝜓 ∨ 𝜒)
Syntax	w3a 1086	Extend wff definition to include three-way conjunction ('and').
wff (𝜑 ∧ 𝜓 ∧ 𝜒)
Definition	df-3or 1087	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.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
Definition	df-3an 1088	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.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
Theorem	3orass 1089	Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒)))
Theorem	3orel1 1090	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 1091	Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒 ∨ 𝜑))
Theorem	3orcoma 1092	Commutation law for triple disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜓 ∨ 𝜑 ∨ 𝜒))
Theorem	3orcomb 1093	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 1094	Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
Theorem	3anan12 1095	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 1097 by Wolf Lammen, 5-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
Theorem	3anan32 1096	Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ 𝜓))
Theorem	3ancoma 1097	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 1098	Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.) (Revised to shorten 3anrot 1099 by Wolf Lammen, 9-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓))
Theorem	3anrot 1099	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 1100	Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓 ∧ 𝜑))