P. 385 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38401-38500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bicontr 38401	Biconditional of its own negation is a contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ ((¬ 𝜑 ↔ 𝜑) ↔ ⊥)
Theorem	impor 38402	An equivalent formula for implying a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∨ 𝜒))
Theorem	orfa 38403	The falsum ⊥ can be removed from a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ ((𝜑 ∨ ⊥) ↔ 𝜑)
Theorem	notbinot2 38404	Commutation rule between negation and biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (¬ (𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ 𝜓))
Theorem	biimpor 38405	A rewriting rule for biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ ((¬ 𝜑 ↔ 𝜓) ∨ 𝜒))
Theorem	orfa1 38406	Add a contradicting disjunct to an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ ⊥) → 𝜓)
Theorem	orfa2 38407	Remove a contradicting disjunct from an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → ⊥)    ⇒   ⊢ ((𝜑 ∨ 𝜓) → 𝜓)
Theorem	bifald 38408	Infer the equivalence to a contradiction from a negation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ↔ ⊥))
Theorem	orsild 38409	A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → ¬ (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	orsird 38410	A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
⊢ (𝜑 → ¬ (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → ¬ 𝜒)
Theorem	cnf1dd 38411	A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → ¬ 𝜒))    &   ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	cnf2dd 38412	A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → ¬ 𝜃))    &   ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	cnfn1dd 38413	A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → (¬ 𝜒 ∨ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	cnfn2dd 38414	A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜑 → (𝜓 → (𝜒 ∨ ¬ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	or32dd 38415	A rearrangement of disjuncts, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜃) ∨ 𝜏)))    ⇒   ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜏) ∨ 𝜃)))
Theorem	notornotel1 38416	A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → ¬ (¬ 𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	notornotel2 38417	A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → ¬ (𝜓 ∨ ¬ 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	contrd 38418	A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
⊢ (𝜑 → (¬ 𝜓 → 𝜒))    &   ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	an12i 38419	An inference from commuting operands in a chain of conjunctions. (Contributed by Giovanni Mascellani, 22-May-2019.)
⊢ (𝜑 ∧ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜓 ∧ (𝜑 ∧ 𝜒))
Theorem	exmid2 38420	An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.)
⊢ ((𝜓 ∧ 𝜑) → 𝜒)    &   ⊢ ((¬ 𝜓 ∧ 𝜂) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜂) → 𝜒)
Theorem	selconj 38421	An inference for selecting one of a list of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ ((𝜂 ∧ 𝜑) ↔ (𝜓 ∧ (𝜂 ∧ 𝜒)))
Theorem	truconj 38422	Add true as a conjunct. (Contributed by Giovanni Mascellani, 23-May-2019.)
⊢ (𝜑 ↔ (⊤ ∧ 𝜑))
Theorem	orel 38423	An inference for disjunction elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ ((𝜓 ∧ 𝜂) → 𝜃)    &   ⊢ ((𝜒 ∧ 𝜌) → 𝜃)    &   ⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ ((𝜑 ∧ (𝜂 ∧ 𝜌)) → 𝜃)
Theorem	negel 38424	An inference for negation elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ (𝜓 → 𝜒)    &   ⊢ (𝜑 → ¬ 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ⊥)
Theorem	botel 38425	An inference for bottom elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ (𝜑 → ⊥)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	tradd 38426	Add top ad a conjunct. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜑 ↔ (⊤ ∧ 𝜓))
Theorem	gm-sbtru 38427	Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ([𝐴 / 𝑥]⊤ ↔ ⊤)
Theorem	sbfal 38428	Substitution does not change falsity. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ([𝐴 / 𝑥]⊥ ↔ ⊥)
Theorem	sbcani 38429	Distribution of class substitution over conjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒)    &   ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂)    ⇒   ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜂))
Theorem	sbcori 38430	Distribution of class substitution over disjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒)    &   ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂)    ⇒   ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ (𝜒 ∨ 𝜂))
Theorem	sbcimi 38431	Distribution of class substitution over implication, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
⊢ 𝐴 ∈ V    &   ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒)    &   ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂)    ⇒   ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜒 → 𝜂))
Theorem	sbcni 38432	Move class substitution inside a negation, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
⊢ 𝐴 ∈ V    &   ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓)
Theorem	sbali 38433	Discard class substitution in a universal quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ([𝐴 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)
Theorem	sbexi 38434	Discard class substitution in an existential quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ([𝐴 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)
Theorem	sbcalf 38435*	Move universal quantifier in and out of class substitution, with an explicit nonfree variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
Theorem	sbcexf 38436*	Move existential quantifier in and out of class substitution, with an explicit nonfree variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
Theorem	sbcalfi 38437*	Move universal quantifier in and out of class substitution, with an explicit nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
⊢ Ⅎ𝑦𝐴    &   ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦𝜓)
Theorem	sbcexfi 38438*	Move existential quantifier in and out of class substitution, with an explicit nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
⊢ Ⅎ𝑦𝐴    &   ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦𝜓)
Theorem	spsbcdi 38439	A lemma for eliminating a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
⊢ 𝐴 ∈ V    &   ⊢ (𝜑 → ∀𝑥𝜒)    &   ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	alrimii 38440*	A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → 𝜓)    &   ⊢ ([𝑦 / 𝑥]𝜒 ↔ 𝜓)    &   ⊢ Ⅎ𝑦𝜒    ⇒   ⊢ (𝜑 → ∀𝑥𝜒)
Theorem	spesbcdi 38441	A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
⊢ (𝜑 → 𝜓)    &   ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜒)
Theorem	exlimddvf 38442	A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.)
⊢ (𝜑 → ∃𝑥𝜃)    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ ((𝜃 ∧ 𝜓) → 𝜒)    &   ⊢ Ⅎ𝑥𝜒    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	exlimddvfi 38443	A lemma for eliminating an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)
⊢ (𝜑 → ∃𝑥𝜃)    &   ⊢ Ⅎ𝑦𝜃    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂)    &   ⊢ ((𝜂 ∧ 𝜓) → 𝜒)    &   ⊢ Ⅎ𝑦𝜒    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	sbceq1ddi 38444	A lemma for eliminating inequality, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝜃)    &   ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜃)    &   ⊢ ([𝐵 / 𝑥]𝜒 ↔ 𝜂)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜂)
Theorem	sbccom2lem 38445*	Lemma for sbccom2 38446. (Contributed by Giovanni Mascellani, 31-May-2019.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Theorem	sbccom2 38446*	Commutative law for double class substitution. (Contributed by Giovanni Mascellani, 31-May-2019.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Theorem	sbccom2f 38447*	Commutative law for double class substitution, with nonfree variable condition. (Contributed by Giovanni Mascellani, 31-May-2019.)
⊢ 𝐴 ∈ V    &   ⊢ Ⅎ𝑦𝐴    ⇒   ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Theorem	sbccom2fi 38448*	Commutative law for double class substitution, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)
⊢ 𝐴 ∈ V    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶    &   ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜓)
Theorem	csbcom2fi 38449*	Commutative law for double class substitution in a class, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)
⊢ 𝐴 ∈ V    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶    &   ⊢ ⦋𝐴 / 𝑥⦌𝐷 = 𝐸    ⇒   ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐸
21.25.2  Tseitin axioms A collection of Tseitin axioms used to convert a wff to Conjunctive Normal Form.
Theorem	fald 38450	Refutation of falsity, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ¬ ⊥)
Theorem	tsim1 38451	A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 → 𝜓)))
Theorem	tsim2 38452	A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜑 ∨ (𝜑 → 𝜓)))
Theorem	tsim3 38453	A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (¬ 𝜓 ∨ (𝜑 → 𝜓)))
Theorem	tsbi1 38454	A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓)))
Theorem	tsbi2 38455	A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓)))
Theorem	tsbi3 38456	A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))
Theorem	tsbi4 38457	A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))
Theorem	tsxo1 38458	A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊻ 𝜓)))
Theorem	tsxo2 38459	A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ⊻ 𝜓)))
Theorem	tsxo3 38460	A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ⊻ 𝜓)))
Theorem	tsxo4 38461	A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ (𝜑 ⊻ 𝜓)))
Theorem	tsan1 38462	A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
Theorem	tsan2 38463	A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜑 ∨ ¬ (𝜑 ∧ 𝜓)))
Theorem	tsan3 38464	A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜓 ∨ ¬ (𝜑 ∧ 𝜓)))
Theorem	tsna1 38465	A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓)))
Theorem	tsna2 38466	A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜑 ∨ (𝜑 ⊼ 𝜓)))
Theorem	tsna3 38467	A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜓 ∨ (𝜑 ⊼ 𝜓)))
Theorem	tsor1 38468	A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
Theorem	tsor2 38469	A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ 𝜑 ∨ (𝜑 ∨ 𝜓)))
Theorem	tsor3 38470	A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ 𝜓 ∨ (𝜑 ∨ 𝜓)))
Theorem	ts3an1 38471	A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → ((¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒) ∨ (𝜑 ∧ 𝜓 ∧ 𝜒)))
Theorem	ts3an2 38472	A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)))
Theorem	ts3an3 38473	A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (𝜒 ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)))
Theorem	ts3or1 38474	A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (((𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ 𝜓 ∨ 𝜒)))
Theorem	ts3or2 38475	A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜓 ∨ 𝜒)))
Theorem	ts3or3 38476	A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ 𝜒 ∨ (𝜑 ∨ 𝜓 ∨ 𝜒)))
21.25.3  Equality deductions A collection of theorems for commuting equalities (or biconditionals) with other constructs.
Theorem	iuneq2f 38477	Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
Theorem	rabeq12f 38478	Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	csbeq12 38479	Equality deduction for substitution in class. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐷)
Theorem	sbeqi 38480	Equality deduction for substitution. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ ((𝑥 = 𝑦 ∧ ∀𝑧(𝜑 ↔ 𝜓)) → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜓))
Theorem	ralbi12f 38481	Equality deduction for restricted universal quantification. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓))
Theorem	oprabbi 38482	Equality deduction for class abstraction of nested ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})
Theorem	mpobi123f 38483*	Equality deduction for maps-to notations with two arguments. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ Ⅎ𝑦𝐷    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ Ⅎ𝑥𝐷    ⇒   ⊢ (((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝐸 = 𝐹) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹))
Theorem	iuneq12f 38484	Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
Theorem	iineq12f 38485	Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)
Theorem	opabbi 38486	Equality deduction for class abstraction of ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Theorem	mptbi12f 38487	Equality deduction for maps-to notations. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → (𝑥 ∈ 𝐴 ↦ 𝐷) = (𝑥 ∈ 𝐵 ↦ 𝐸))
21.25.4  Miscellanea Work in progress or things that do not belong anywhere else.
Theorem	orcomdd 38488	Commutativity of logic disjunction, in double deduction form. Should not be moved to main, see PR #3034 in Github. Use orcomd 872 instead. (Contributed by Giovanni Mascellani, 19-Mar-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 ∨ 𝜒)))
Theorem	scottexf 38489*	A version of scottex 9809 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
Theorem	scott0f 38490*	A version of scott0 9810 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Theorem	scottn0f 38491*	A version of scott0f 38490 with inequalities instead of equalities. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ≠ ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)
Theorem	ac6s3f 38492*	Generalization of the Axiom of Choice to classes, with bound-variable hypothesis. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 𝜓)
Theorem	ac6s6 38493*	Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)
Theorem	ac6s6f 38494*	Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 20-Aug-2018.)
⊢ 𝐴 ∈ V    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)
21.26  Mathbox for Peter Mazsa
21.26.1  Notations
Syntax	cxrn 38495	Extend the definition of a class to include the range Cartesian product class.
class (𝐴 ⋉ 𝐵)
Syntax	cqmap 38496	Extend the definition of a class to include the quotient map of a class.
class QMap 𝑅
Syntax	cadjliftmap 38497	Extend the definition of a class to include the class of adjoined lift maps.
class (𝑅 AdjLiftMap 𝐴)
Syntax	cblockliftmap 38498	Extend the definition of a class to include the class of block lift maps.
class (𝑅 BlockLiftMap 𝐴)
Syntax	csucmap 38499	Extend the definition of a class to include the class of successor maps.
class SucMap
Syntax	csuccl 38500	Extend the definition of a class to include the class of successors.
class Suc