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	orimdidc 901	Disjunction distributes over implication. The forward direction, pm2.76 803, is valid intuitionistically. The reverse direction holds if 𝜑 is decidable, as can be seen at pm2.85dc 900. (Contributed by Jim Kingdon, 1-Apr-2018.)
⊢ (DECID 𝜑 → ((𝜑 ∨ (𝜓 → 𝜒)) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒))))
Theorem	pm2.26dc 902	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 903	Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 702 which holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.)
⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) ↔ 𝜑))
Theorem	pm5.11dc 904	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 905	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 906	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 907	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 908	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 909	Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 831, condc 848, or notnotrdc 838 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 910	The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 890, 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 911	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	pm5.35 912	Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒)))
Theorem	pm5.54dc 913	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 914	Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜓 → (𝜑 ↔ 𝜒))
Theorem	baibr 915	Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜓 → (𝜒 ↔ 𝜑))
Theorem	rbaib 916	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜒 → (𝜑 ↔ 𝜓))
Theorem	rbaibr 917	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜒 → (𝜓 ↔ 𝜑))
Theorem	baibd 918	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃))
Theorem	rbaibd 919	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜒))
Theorem	pm5.44 920	Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) → ((𝜑 → 𝜒) ↔ (𝜑 → (𝜓 ∧ 𝜒))))
Theorem	pm5.6dc 921	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 922). (Contributed by Jim Kingdon, 2-Apr-2018.)
⊢ (DECID 𝜓 → (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒))))
Theorem	pm5.6r 922	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 921). (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ ((𝜑 → (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))
Theorem	orcanai 923	Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.)
⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒)
Theorem	intnan 924	Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.)
⊢ ¬ 𝜑    ⇒   ⊢ ¬ (𝜓 ∧ 𝜑)
Theorem	intnanr 925	Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.)
⊢ ¬ 𝜑    ⇒   ⊢ ¬ (𝜑 ∧ 𝜓)
Theorem	intnand 926	Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ (𝜒 ∧ 𝜓))
Theorem	intnanrd 927	Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒))
Theorem	dcan 928	A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∧ 𝜓))
Theorem	dcan2 929	A conjunction of two decidable propositions is decidable, expressed in a curried form as compared to dcan 928. (Contributed by Jim Kingdon, 12-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∧ 𝜓)))
Theorem	dcor 930	A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
Theorem	dcbi 931	An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ↔ 𝜓)))
Theorem	annimdc 932	Express conjunction in terms of implication. The forward direction, annimim 681, 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 933	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 934	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 935	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 936	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 937	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 938	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 939	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 940	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 941	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 942	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 943	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	pm4.43 944	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	pm4.82 945	Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑)
Theorem	pm4.83dc 946	Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 860, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)
⊢ (DECID 𝜑 → (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)) ↔ 𝜓))
Theorem	biantr 947	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 948	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 949	Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 948. (Contributed by Jim Kingdon, 2-Apr-2018.)
⊢ (DECID 𝜒 → (((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)) ↔ (𝜒 ∨ (𝜑 ↔ 𝜓))))
Theorem	bigolden 950	Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)
⊢ (((𝜑 ∧ 𝜓) ↔ 𝜑) ↔ (𝜓 ↔ (𝜑 ∨ 𝜓)))
Theorem	anordc 951	Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 749, 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 952	Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 749, 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 953	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 954	Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 748, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))))
Theorem	dn1dc 955	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 956	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	pm5.75 957	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 958	Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)
⊢ (((𝜑 → 𝜓) ∧ (𝜒 ↔ (𝜓 ∧ 𝜑))) → (𝜒 ↔ 𝜑))
Theorem	ccase 959	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 960	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 961	Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
⊢ ((𝜑 ∧ 𝜓) → 𝜏)    &   ⊢ (𝜒 → 𝜏)    &   ⊢ (𝜃 → 𝜏)    ⇒   ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏)
Theorem	niabn 962	Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜓 → ((𝜒 ∧ 𝜓) ↔ ¬ 𝜑))
Theorem	dedlem0a 963	Alternate version of dedlema 964. (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 964	Lemma for iftrue 3531. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ (𝜑 → (𝜓 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Theorem	dedlemb 965	Lemma for iffalse 3534. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
Theorem	pm4.42r 966	One direction of Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓)) → 𝜑)
Theorem	ninba 967	Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
⊢ 𝜑    ⇒   ⊢ (¬ 𝜓 → (¬ 𝜑 ↔ (𝜒 ∧ 𝜓)))
Theorem	prlem1 968	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 969	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 970	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 971	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 972	Extend wff definition to include 3-way disjunction ('or').
wff (𝜑 ∨ 𝜓 ∨ 𝜒)
Syntax	w3a 973	Extend wff definition to include 3-way conjunction ('and').
wff (𝜑 ∧ 𝜓 ∧ 𝜒)
Definition	df-3or 974	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 762. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
Definition	df-3an 975	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 399. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
Theorem	3orass 976	Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜑 ∨ (𝜓 ∨ 𝜒)))
Theorem	3anass 977	Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
Theorem	3anrot 978	Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒 ∧ 𝜑))
Theorem	3orrot 979	Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒 ∨ 𝜑))
Theorem	3ancoma 980	Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒))
Theorem	3ancomb 981	Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓))
Theorem	3orcomb 982	Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (𝜑 ∨ 𝜒 ∨ 𝜓))
Theorem	3anrev 983	Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓 ∧ 𝜑))
Theorem	3anan32 984	Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ 𝜓))
Theorem	3anan12 985	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 986	Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
Theorem	anandi3r 987	Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓)))
Theorem	3ioran 988	Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
Theorem	3simpa 989	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜓))
Theorem	3simpb 990	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒))
Theorem	3simpc 991	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 992	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜑)
Theorem	simp2 993	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓)
Theorem	simp3 994	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜒)
Theorem	simpl1 995	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜑)
Theorem	simpl2 996	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜓)
Theorem	simpl3 997	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜒)
Theorem	simpr1 998	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜓)
Theorem	simpr2 999	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜒)
Theorem	simpr3 1000	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜃)