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	pm5.62 1001	Theorem *5.62 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 21-Jun-2005.)
⊢ (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))
Theorem	pm5.63 1002	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 1003	Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜓 → ((𝜒 ∧ 𝜓) ↔ ¬ 𝜑))
Theorem	ninba 1004	Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜓 → (¬ 𝜑 ↔ (𝜒 ∧ 𝜓)))
Theorem	pm4.43 1005	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 1006	Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)
Theorem	pm4.83 1007	Theorem *4.83 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)) ↔ 𝜓)
Theorem	pclem6 1008	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 1009	Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)
⊢ (((𝜑 ∧ 𝜓) ↔ 𝜑) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓)))
Theorem	pm5.71 1010	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 1011	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 1012	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 1013	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 1014	Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)
⊢ (¬ 𝜑 → 𝜒)    &   ⊢ (¬ 𝜓 → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ 𝜒
Theorem	ecase3d 1015	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 1016	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 1017	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 1018	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 1019	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 1020	Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
⊢ ((𝜑 ∧ 𝜓) → 𝜏)    &   ⊢ (𝜒 → 𝜏)    &   ⊢ (𝜃 → 𝜏)    ⇒   ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏)
Theorem	4cases 1021	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 1022	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 1023	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 1024	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 1025	Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.)
⊢ (¬ 𝜑 → (𝜓 ↔ ((𝜓 → 𝜑) → (𝜒 ∧ 𝜑))))
Theorem	dedlema 1026	Lemma for weak deduction theorem. See also ifptru 1054. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ (𝜑 → (𝜓 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Theorem	dedlemb 1027	Lemma for weak deduction theorem. See also ifpfal 1055. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Theorem	cases2 1028	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 1029	Alternate proof of cases2 1028, not using dedlema 1026 or dedlemb 1027. (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 1030	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 1031	Theorem *5.24 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
Theorem	4exmid 1032	The disjunction of the four possible combinations of two wffs and their negations is always true. A four-way excluded middle (see exmid 878). (Contributed by David Abernethy, 28-Jan-2014.) (Proof shortened by NM, 29-Oct-2021.)
⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
Theorem	consensus 1033	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 1034	Theorem *4.42 of [WhiteheadRussell] p. 119. See also ifpid 1056. (Contributed by Roy F. Longton, 21-Jun-2005.)
⊢ (𝜑 ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))
Theorem	prlem1 1035	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 1036	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 1037	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 1038	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 1039	A closed form of mpbir 223, 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 1040	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 1041	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 1042	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 1044). It is the analogue for propositions of the conditional operator for classes, denoted by if(𝜑, 𝐴, 𝐵) (see df-if 4345).
Syntax	wif 1043	Extend class notation to include the conditional operator for propositions.
wff if-(𝜑, 𝜓, 𝜒)
Definition	df-ifp 1044	Definition of the conditional operator for propositions. The expression if-(𝜑, 𝜓, 𝜒) is read "if 𝜑 then 𝜓 else 𝜒". See dfifp2 1045, dfifp3 1046, dfifp4 1047, dfifp5 1048, dfifp6 1049 and dfifp7 1050 for alternate definitions. This definition (in the form of dfifp2 1045) 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 1045 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 1045	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 1044). (Contributed by BJ, 22-Jun-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
Theorem	dfifp3 1046	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
Theorem	dfifp4 1047	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
Theorem	dfifp5 1048	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 → 𝜒)))
Theorem	dfifp6 1049	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜒 → 𝜑)))
Theorem	dfifp7 1050	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒 → 𝜑) → (𝜑 ∧ 𝜓)))
Theorem	anifp 1051	The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1052. (Contributed by BJ, 30-Sep-2019.)
⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
Theorem	ifpor 1052	The conditional operator implies the disjunction of its possible outputs. Dual statement of anifp 1051. (Contributed by BJ, 1-Oct-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) → (𝜓 ∨ 𝜒))
Theorem	ifpn 1053	Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
Theorem	ifptru 1054	Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4350. This is essentially dedlema 1026. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))
Theorem	ifpfal 1055	Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4353. This is essentially dedlemb 1027. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 25-Jun-2020.)
⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))
Theorem	ifpid 1056	Value of the conditional operator for propositions when the same proposition is returned in either case. Analogue for propositions of ifid 4383. This is essentially pm4.42 1034. (Contributed by BJ, 20-Sep-2019.)
⊢ (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓)
Theorem	casesifp 1057	Version of cases 1023 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 1054 and ifpfal 1055. (Contributed by BJ, 20-Sep-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (¬ 𝜑 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜓 ↔ if-(𝜑, 𝜒, 𝜃))
Theorem	ifpbi123d 1058	Equality deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜏))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜂))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜁))    ⇒   ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))
Theorem	ifpimpda 1059	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 1060	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 4400. For more information on the weak deduction theorem, see the Weak Deduction Theorem page mmdeduction.html.
Theorem	elimh 1061	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 1062	Obsolete version of elimh 1061 as of 27-Apr-2023. 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.) (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 1063	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 1064	Obsolete version of dedt 1063 as of 27-Apr-2023. 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.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜃 ↔ 𝜏))    &   ⊢ 𝜏    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	con3ALT 1065	Proof of con3 151 from its associated inference con3i 152 that illustrates the use of the weak deduction theorem dedt 1063. (Contributed by NM, 27-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) Revised dedt 1063 and elimh 1061. (Revised by Steven Nguyen, 27-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
Theorem	con3ALTOLD 1066	Obsolete version of con3ALT 1065 as of 27-Apr-2023. Proof of con3 151 from its associated inference con3i 152 that illustrates the use of the weak deduction theorem dedt 1063. (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))
1.2.11  Abbreviated conjunction and disjunction of three wff's
Syntax	w3o 1067	Extend wff definition to include three-way disjunction ('or').
wff (𝜑 ∨ 𝜓 ∨ 𝜒)
Syntax	w3a 1068	Extend wff definition to include three-way conjunction ('and').
wff (𝜑 ∧ 𝜓 ∧ 𝜒)
Definition	df-3or 1069	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 905. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
Definition	df-3an 1070	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 461. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
Theorem	3orass 1071	Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒)))
Theorem	3orel1 1072	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 1073	Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒 ∨ 𝜑))
Theorem	3orcoma 1074	Commutation law for triple disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜓 ∨ 𝜑 ∨ 𝜒))
Theorem	3orcomb 1075	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 1076	Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
Theorem	3anan12 1077	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 1079 by Wolf Lammen, 5-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
Theorem	3anan32 1078	Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ 𝜓))
Theorem	3ancoma 1079	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 1080	Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.) (Revised to shorten 3anrot 1081 by Wolf Lammen, 9-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓))
Theorem	3anrot 1081	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 1082	Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓 ∧ 𝜑))
Theorem	anandi3 1083	Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
Theorem	anandi3r 1084	Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓)))
Theorem	3an4anass 1085	Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
Theorem	3ioran 1086	Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
Theorem	3ianor 1087	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.)
⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
Theorem	3anor 1088	Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Wolf Lammen, 8-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
Theorem	3oran 1089	Triple disjunction in terms of triple conjunction. (Contributed by NM, 8-Oct-2012.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
Theorem	3impa 1090	Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.) (Revised to shorten 3imp 1091 by Wolf Lammen, 20-Jun-2022.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3imp 1091	Importation inference. (Contributed by NM, 8-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jun-2022.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3imp31 1092	The importation inference 3imp 1091 with commutation of the first and third conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜑) → 𝜃)
Theorem	3imp231 1093	Importation inference. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → 𝜃)
Theorem	3imp21 1094	The importation inference 3imp 1091 with commutation of the first and second conjuncts of the assertion relative to the hypothesis. (Contributed by Alan Sare, 11-Sep-2016.) (Revised to shorten 3com12 1103 by Wolf Lammen, 23-Jun-2022.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃)
Theorem	3impb 1095	Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3impib 1096	Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3impia 1097	Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.) (Proof shortened by Wolf Lammen, 21-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	3expa 1098	Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.) (Revised to shorten 3exp 1099 and pm3.2an3 1320 by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
Theorem	3exp 1099	Exportation inference. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 22-Jun-2022.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
Theorem	3expb 1100	Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)