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	ecase3 1001	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 1002	Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)
⊢ (¬ 𝜑 → 𝜒)    &   ⊢ (¬ 𝜓 → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ 𝜒
Theorem	ecase3d 1003	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 1004	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 1005	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 1006	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 1007	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 1008	Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
⊢ ((𝜑 ∧ 𝜓) → 𝜏)    &   ⊢ (𝜒 → 𝜏)    &   ⊢ (𝜃 → 𝜏)    ⇒   ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏)
Theorem	4cases 1009	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 1010	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 1011	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 1012	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 1013	Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.)
⊢ (¬ 𝜑 → (𝜓 ↔ ((𝜓 → 𝜑) → (𝜒 ∧ 𝜑))))
Theorem	dedlema 1014	Lemma for weak deduction theorem. See also ifptru 1043. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ (𝜑 → (𝜓 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Theorem	dedlemb 1015	Lemma for weak deduction theorem. See also ifpfal 1044. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Theorem	cases2 1016	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 1017	Alternate proof of cases2 1016, not using dedlema 1014 or dedlemb 1015. (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 1018	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	dfbi3OLD 1019	Obsolete proof of dfbi3 1018 as of 29-Oct-2021. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
Theorem	pm5.24 1020	Theorem *5.24 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)
⊢ (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
Theorem	4exmid 1021	The disjunction of the four possible combinations of two wffs and their negations is always true. A four-way excluded middle (see exmid 430). (Contributed by David Abernethy, 28-Jan-2014.) (Proof shortened by NM, 29-Oct-2021.)
⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
Theorem	4exmidOLD 1022	Obsolete proof of 4exmid 1021 as of 29-Oct-2021. (Contributed by David Abernethy, 28-Jan-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
Theorem	consensus 1023	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 1024	Theorem *4.42 of [WhiteheadRussell] p. 119. See also ifpid 1045. (Contributed by Roy F. Longton, 21-Jun-2005.)
⊢ (𝜑 ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))
Theorem	prlem1 1025	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 1026	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 1027	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 1028	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 1029	A closed form of mpbir 221, 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 1030	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 1031	Inference separating a disjunct of an antecedent. (Contributed by Alexander van der Vekens, 25-May-2018.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((¬ 𝜑 ∧ 𝜒) → 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜒) → 𝜓)
1.2.8  The conditional operator for propositions This subsection introduces the conditional operator for propositions, denoted by if- (see df-ifp 1033). It is the analogue for propositions of the conditional operator for classes if (see df-if 4120).
Syntax	wif 1032	Extend class notation to include the conditional operator for propositions.
wff if-(𝜑, 𝜓, 𝜒)
Definition	df-ifp 1033	Definition of the conditional operator for propositions. The value of if-(𝜑, 𝜓, 𝜒) is 𝜓 if 𝜑 is true and 𝜒 if 𝜑 false. See dfifp2 1034, dfifp3 1035, dfifp4 1036, dfifp5 1037, dfifp6 1038 and dfifp7 1039 for alternate definitions. This definition (in the form of dfifp2 1034) 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). 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 wff with n+1 variables; single out one variable, say 𝜑; when one sets 𝜑 to True (resp. False), then what remains is a wff of n variables, so by the induction hypothesis it corresponds to a formula using only {if-, ⊤, ⊥}, say 𝜓 (resp. 𝜒); therefore, the formula if-(𝜑, 𝜓, 𝜒) represents the initial wff. Now, since { → , ¬ } and similar systems suffice to express if-, ⊤, ⊥, they are also complete. (Contributed by BJ, 22-Jun-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
Theorem	dfifp2 1034	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 1033). (Contributed by BJ, 22-Jun-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
Theorem	dfifp3 1035	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)))
Theorem	dfifp4 1036	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
Theorem	dfifp5 1037	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 → 𝜒)))
Theorem	dfifp6 1038	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜒 → 𝜑)))
Theorem	dfifp7 1039	Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒 → 𝜑) → (𝜑 ∧ 𝜓)))
Theorem	anifp 1040	The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1041. (Contributed by BJ, 30-Sep-2019.)
⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
Theorem	ifpor 1041	The conditional operator implies the disjunction of its possible outputs. Dual statement of anifp 1040. (Contributed by BJ, 1-Oct-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) → (𝜓 ∨ 𝜒))
Theorem	ifpn 1042	Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.)
⊢ (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
Theorem	ifptru 1043	Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4125. This is essentially dedlema 1014. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
⊢ (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))
Theorem	ifpfal 1044	Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4128. This is essentially dedlemb 1015. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 25-Jun-2020.)
⊢ (¬ 𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))
Theorem	ifpid 1045	Value of the conditional operator for propositions when the same proposition is returned in either case. Analogue for propositions of ifid 4158. This is essentially pm4.42 1024. (Contributed by BJ, 20-Sep-2019.)
⊢ (if-(𝜑, 𝜓, 𝜓) ↔ 𝜓)
Theorem	casesifp 1046	Version of cases 1011 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 1043 and ifpfal 1044. (Contributed by BJ, 20-Sep-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (¬ 𝜑 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜓 ↔ if-(𝜑, 𝜒, 𝜃))
Theorem	ifpbi123d 1047	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 1048	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 1049	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)
1.2.9  The weak deduction theorem This subsection contains a few results related to the weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html.
Theorem	elimh 1050	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.)
⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜒 ↔ 𝜏))    &   ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜓) → (𝜃 ↔ 𝜏))    &   ⊢ 𝜃    ⇒   ⊢ 𝜏
Theorem	dedt 1051	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.)
⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜃 ↔ 𝜏))    &   ⊢ 𝜏    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	con3ALT 1052	Proof of con3 149 from its associated inference con3i 150 that illustrates the use of the weak deduction theorem dedt 1051. (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))
1.2.10  Abbreviated conjunction and disjunction of three wff's
Syntax	w3o 1053	Extend wff definition to include three-way disjunction ('or').
wff (𝜑 ∨ 𝜓 ∨ 𝜒)
Syntax	w3a 1054	Extend wff definition to include three-way conjunction ('and').
wff (𝜑 ∧ 𝜓 ∧ 𝜒)
Definition	df-3or 1055	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 545. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
Definition	df-3an 1056	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 682. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
Theorem	3orass 1057	Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒)))
Theorem	3orel1 1058	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	3anass 1059	Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
Theorem	3anrot 1060	Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒 ∧ 𝜑))
Theorem	3orrot 1061	Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒 ∨ 𝜑))
Theorem	3ancoma 1062	Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒))
Theorem	3orcoma 1063	Commutation law for triple disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜓 ∨ 𝜑 ∨ 𝜒))
Theorem	3ancomb 1064	Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓))
Theorem	3orcomb 1065	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	3orcombOLD 1066	Obsolete version of 3orcomb 1065 as of 8-Apr-2022. (Contributed by Scott Fenton, 20-Apr-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜑 ∨ 𝜒 ∨ 𝜓))
Theorem	3anrev 1067	Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓 ∧ 𝜑))
Theorem	3anan32 1068	Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ 𝜓))
Theorem	3anan12 1069	Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
Theorem	anandi3 1070	Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
Theorem	anandi3r 1071	Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓)))
Theorem	3anorOLD 1072	Obsolete version of 3anor 1075 as of 8-Apr-2022. (Contributed by Jeff Hankins, 15-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
Theorem	3ioran 1073	Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
Theorem	3ianor 1074	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 1075	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	3ianorOLD 1076	Obsolete version of 3ianor 1074 as of 8-Apr-2022. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
Theorem	3oran 1077	Triple disjunction in terms of triple conjunction. (Contributed by NM, 8-Oct-2012.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
Theorem	3simpa 1078	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜓))
Theorem	3simpb 1079	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒))
Theorem	3simpc 1080	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒))
Theorem	simp1 1081	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜑)
Theorem	simp2 1082	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓)
Theorem	simp3 1083	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜒)
Theorem	simpl1 1084	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜑)
Theorem	simpl2 1085	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜓)
Theorem	simpl3 1086	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜒)
Theorem	simpr1 1087	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜓)
Theorem	simpr2 1088	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜒)
Theorem	simpr3 1089	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜃)
Theorem	simp1i 1090	Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒)    ⇒   ⊢ 𝜑
Theorem	simp2i 1091	Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒)    ⇒   ⊢ 𝜓
Theorem	simp3i 1092	Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
⊢ (𝜑 ∧ 𝜓 ∧ 𝜒)    ⇒   ⊢ 𝜒
Theorem	simp1d 1093	Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	simp2d 1094	Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	simp3d 1095	Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	simp1bi 1096	Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	simp2bi 1097	Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	simp3bi 1098	Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	3adant1 1099	Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜃 ∧ 𝜑 ∧ 𝜓) → 𝜒)
Theorem	3adant2 1100	Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜒)