P. 10 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 901-1000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pm4.64dc 901	Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 723, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.)
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓)))
Theorem	pm4.66dc 902	Theorem *4.66 of [WhiteheadRussell] p. 120, given a decidability condition. (Contributed by Jim Kingdon, 2-May-2018.)
⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
Theorem	pm4.54dc 903	Theorem *4.54 of [WhiteheadRussell] p. 120, for decidable propositions. One form of DeMorgan's law. (Contributed by Jim Kingdon, 2-May-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))
Theorem	pm4.79dc 904	Equivalence between a disjunction of two implications, and a conjunction and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121 but with additional decidability antecedents. (Contributed by Jim Kingdon, 28-Mar-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → (((𝜓 → 𝜑) ∨ (𝜒 → 𝜑)) ↔ ((𝜓 ∧ 𝜒) → 𝜑))))
Theorem	pm5.17dc 905	Two ways of stating exclusive-or which are equivalent for a decidable proposition. Based on theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 16-Apr-2018.)
⊢ (DECID 𝜓 → (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)))
Theorem	pm2.85dc 906	Reverse distribution of disjunction over implication, given decidability. Based on theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by Jim Kingdon, 1-Apr-2018.)
⊢ (DECID 𝜑 → (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 → 𝜒))))
Theorem	orimdidc 907	Disjunction distributes over implication. The forward direction, pm2.76 809, is valid intuitionistically. The reverse direction holds if 𝜑 is decidable, as can be seen at pm2.85dc 906. (Contributed by Jim Kingdon, 1-Apr-2018.)
⊢ (DECID 𝜑 → ((𝜑 ∨ (𝜓 → 𝜒)) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒))))
Theorem	pm2.26dc 908	Decidable proposition version of theorem *2.26 of [WhiteheadRussell] p. 104. (Contributed by Jim Kingdon, 20-Apr-2018.)
⊢ (DECID 𝜑 → (¬ 𝜑 ∨ ((𝜑 → 𝜓) → 𝜓)))
Theorem	pm4.81dc 909	Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 708 which holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.)
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) ↔ 𝜑))
Theorem	pm5.11dc 910	A decidable proposition or its negation implies a second proposition. Based on theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 29-Mar-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 → 𝜓) ∨ (¬ 𝜑 → 𝜓))))
Theorem	pm5.12dc 911	Excluded middle with antecedents for a decidable consequent. Based on theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ∨ (𝜑 → ¬ 𝜓)))
Theorem	pm5.14dc 912	A decidable proposition is implied by or implies other propositions. Based on theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ∨ (𝜓 → 𝜒)))
Theorem	pm5.13dc 913	An implication holds in at least one direction, where one proposition is decidable. Based on theorem *5.13 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ∨ (𝜓 → 𝜑)))
Theorem	pm5.55dc 914	A disjunction is equivalent to one of its disjuncts, given a decidable disjunct. Based on theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)
⊢ (DECID 𝜑 → (((𝜑 ∨ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∨ 𝜓) ↔ 𝜓)))
Theorem	peircedc 915	Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 837, condc 854, or notnotrdc 844 in the sense of expressing the "difference" between an intuitionistic system of propositional calculus and a classical system. In intuitionistic logic, it only holds for decidable propositions. (Contributed by Jim Kingdon, 3-Jul-2018.)
⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
Theorem	looinvdc 916	The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 896, we can see that this expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108 (plus the decidability condition). (Contributed by NM, 12-Aug-2004.)
⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → ((𝜓 → 𝜑) → 𝜑)))
1.2.10  Miscellaneous theorems of propositional calculus
Theorem	pm5.21nd 917	Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    &   ⊢ (𝜃 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	pm5.35 918	Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒)))
Theorem	pm5.54dc 919	A conjunction is equivalent to one of its conjuncts, given a decidable conjunct. Based on theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)
⊢ (DECID 𝜑 → (((𝜑 ∧ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∧ 𝜓) ↔ 𝜓)))
Theorem	baib 920	Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜓 → (𝜑 ↔ 𝜒))
Theorem	baibr 921	Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜓 → (𝜒 ↔ 𝜑))
Theorem	rbaib 922	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜒 → (𝜑 ↔ 𝜓))
Theorem	rbaibr 923	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜒 → (𝜓 ↔ 𝜑))
Theorem	baibd 924	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃))
Theorem	rbaibd 925	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜒))
Theorem	pm5.44 926	Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) → ((𝜑 → 𝜒) ↔ (𝜑 → (𝜓 ∧ 𝜒))))
Theorem	pm5.6dc 927	Conjunction in antecedent versus disjunction in consequent, for a decidable proposition. Theorem *5.6 of [WhiteheadRussell] p. 125, with decidability condition added. The reverse implication holds for all propositions (see pm5.6r 928). (Contributed by Jim Kingdon, 2-Apr-2018.)
⊢ (DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒))))
Theorem	pm5.6r 928	Conjunction in antecedent versus disjunction in consequent. One direction of Theorem *5.6 of [WhiteheadRussell] p. 125. If 𝜓 is decidable, the converse also holds (see pm5.6dc 927). (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ ((𝜑 → (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))
Theorem	orcanai 929	Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒)
Theorem	intnan 930	Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.)
⊢ ¬ 𝜑    ⇒   ⊢ ¬ (𝜓 ∧ 𝜑)
Theorem	intnanr 931	Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.)
⊢ ¬ 𝜑    ⇒   ⊢ ¬ (𝜑 ∧ 𝜓)
Theorem	intnand 932	Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ (𝜒 ∧ 𝜓))
Theorem	intnanrd 933	Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒))
Theorem	dcand 934	A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.) (Revised by BJ, 14-Nov-2024.)
⊢ (𝜑 → DECID 𝜓)    &   ⊢ (𝜑 → DECID 𝜒)    ⇒   ⊢ (𝜑 → DECID (𝜓 ∧ 𝜒))
Theorem	dcan 935	A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∧ 𝜓))
Theorem	dcan2 936	A conjunction of two decidable propositions is decidable, expressed in a curried form as compared to dcan 935. This is deprecated; it's trivial to recreate with ex 115, but it's here in case someone is using this older form. (Contributed by Jim Kingdon, 12-Apr-2018.) (New usage is discouraged.)
⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∧ 𝜓)))
Theorem	dcor 937	A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
Theorem	dcbi 938	An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ↔ 𝜓)))
Theorem	annimdc 939	Express conjunction in terms of implication. The forward direction, annimim 687, is valid for all propositions, but as an equivalence, it requires a decidability condition. (Contributed by Jim Kingdon, 25-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))))
Theorem	pm4.55dc 940	Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∧ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))))
Theorem	orandc 941	Disjunction in terms of conjunction (De Morgan's law), for decidable propositions. Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by Jim Kingdon, 13-Dec-2021.)
⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∨ 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))
Theorem	mpbiran 942	Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.) (Revised by NM, 9-Jan-2015.)
⊢ 𝜓    &   ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 ↔ 𝜒)
Theorem	mpbiran2 943	Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.) (Revised by NM, 9-Jan-2015.)
⊢ 𝜒    &   ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	mpbir2an 944	Detach a conjunction of truths in a biconditional. (Contributed by NM, 10-May-2005.) (Revised by NM, 9-Jan-2015.)
⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ 𝜑
Theorem	mpbi2and 945	Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 𝜃))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	mpbir2and 946	Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm5.62dc 947	Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
⊢ (DECID 𝜓 → (((𝜑 ∧ 𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
Theorem	pm5.63dc 948	Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ (¬ 𝜑 ∧ 𝜓))))
Theorem	bianfi 949	A wff conjoined with falsehood is false. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)
⊢ ¬ 𝜑    ⇒   ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑))
Theorem	bianfd 950	A wff conjoined with falsehood is false. (Contributed by NM, 27-Mar-1995.) (Proof shortened by Wolf Lammen, 5-Nov-2013.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒)))
Theorem	pm4.43 951	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	pm4.82 952	Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)
Theorem	pm4.83dc 953	Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 866, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)
⊢ (DECID 𝜑 → (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)) ↔ 𝜓))
Theorem	biantr 954	A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜓)) → (𝜑 ↔ 𝜒))
Theorem	orbididc 955	Disjunction distributes over the biconditional, for a decidable proposition. Based on an axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384. (Contributed by Jim Kingdon, 2-Apr-2018.)
⊢ (DECID 𝜑 → ((𝜑 ∨ (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∨ 𝜒))))
Theorem	pm5.7dc 956	Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 955. (Contributed by Jim Kingdon, 2-Apr-2018.)
⊢ (DECID 𝜒 → (((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)) ↔ (𝜒 ∨ (𝜑 ↔ 𝜓))))
Theorem	bigolden 957	Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)
⊢ (((𝜑 ∧ 𝜓) ↔ 𝜑) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓)))
Theorem	anordc 958	Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 755, holds for all propositions, but the equivalence only holds given decidability. (Contributed by Jim Kingdon, 21-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))
Theorem	pm3.11dc 959	Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 755, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))))
Theorem	pm3.12dc 960	Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓))))
Theorem	pm3.13dc 961	Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 754, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))))
Theorem	dn1dc 962	DN1 for decidable propositions. Without the decidability conditions, DN1 can serve as a single axiom for Boolean algebra. See http://www-unix.mcs.anl.gov/~mccune/papers/basax/v12.pdf. (Contributed by Jim Kingdon, 22-Apr-2018.)
⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → (¬ (¬ (¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))) ↔ 𝜒))
Theorem	pm5.71dc 963	Decidable proposition version of theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.) (Modified for decidability by Jim Kingdon, 19-Apr-2018.)
⊢ (DECID 𝜓 → ((𝜓 → ¬ 𝜒) → (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ 𝜒))))
Theorem	pm5.75 964	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.)
⊢ (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓 ∨ 𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))
Theorem	bimsc1 965	Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)
⊢ (((𝜑 → 𝜓) ∧ (𝜒 ↔ (𝜓 ∧ 𝜑))) → (𝜒 ↔ 𝜑))
Theorem	ccase 966	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 967	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 968	Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
⊢ ((𝜑 ∧ 𝜓) → 𝜏)    &   ⊢ (𝜒 → 𝜏)    &   ⊢ (𝜃 → 𝜏)    ⇒   ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏)
Theorem	niabn 969	Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜓 → ((𝜒 ∧ 𝜓) ↔ ¬ 𝜑))
Theorem	dedlem0a 970	Alternate version of dedlema 971. (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	dedlema 971	Lemma for iftrue 3554. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ (𝜑 → (𝜓 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Theorem	dedlemb 972	Lemma for iffalse 3557. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Theorem	pm4.42r 973	One direction of Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓)) → 𝜑)
Theorem	ninba 974	Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜓 → (¬ 𝜑 ↔ (𝜒 ∧ 𝜓)))
Theorem	prlem1 975	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 976	A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 5-Aug-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 977	A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    &   ⊢ (𝜑 → (𝜃 ∨ 𝜏))    &   ⊢ (𝜓 ↔ 𝜃)    &   ⊢ (𝜒 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	rnlem 978	Lemma used in construction of real numbers. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ∧ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒))))
1.2.11  Abbreviated conjunction and disjunction of three wff's
Syntax	w3o 979	Extend wff definition to include 3-way disjunction ('or').
wff (𝜑 ∨ 𝜓 ∨ 𝜒)
Syntax	w3a 980	Extend wff definition to include 3-way conjunction ('and').
wff (𝜑 ∧ 𝜓 ∧ 𝜒)
Definition	df-3or 981	Define disjunction ('or') of 3 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 768. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
Definition	df-3an 982	Define conjunction ('and') of 3 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 401. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
Theorem	3orass 983	Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒)))
Theorem	3anass 984	Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
Theorem	3anrot 985	Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒 ∧ 𝜑))
Theorem	3orrot 986	Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒 ∨ 𝜑))
Theorem	3ancoma 987	Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒))
Theorem	3ancomb 988	Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓))
Theorem	3orcomb 989	Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜑 ∨ 𝜒 ∨ 𝜓))
Theorem	3anrev 990	Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓 ∧ 𝜑))
Theorem	3anan32 991	Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ 𝜓))
Theorem	3anan12 992	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 993	Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
Theorem	anandi3r 994	Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓)))
Theorem	3ioran 995	Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
Theorem	3simpa 996	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜓))
Theorem	3simpb 997	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒))
Theorem	3simpc 998	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 999	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜑)
Theorem	simp2 1000	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓)