P. 391 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39001-39100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	truconj 39001	Add true as a conjunct. (Contributed by Giovanni Mascellani, 23-May-2019.)
⊢ (𝜑 ↔ (⊤ ∧ 𝜑))
Theorem	orel 39002	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 39003	An inference for negation elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ (𝜓 → 𝜒)    &   ⊢ (𝜑 → ¬ 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ⊥)
Theorem	botel 39004	An inference for bottom elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ (𝜑 → ⊥)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	tradd 39005	Add top ad a conjunct. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜑 ↔ (⊤ ∧ 𝜓))
Theorem	gm-sbtru 39006	Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ([𝐴 / 𝑥]⊤ ↔ ⊤)
Theorem	sbfal 39007	Substitution does not change falsity. (Contributed by Giovanni Mascellani, 24-May-2019.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ([𝐴 / 𝑥]⊥ ↔ ⊥)
Theorem	sbcani 39008	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 39009	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 39010	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 39011	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 39012	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 39013	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 39014*	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 39015*	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 39016*	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 39017*	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 39018	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 39019*	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 39020	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 39021	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 39022	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 39023	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 39024*	Lemma for sbccom2 39025. (Contributed by Giovanni Mascellani, 31-May-2019.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Theorem	sbccom2 39025*	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 39026*	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 39027*	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 39028*	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
21.26.2  Tseitin axioms A collection of Tseitin axioms used to convert a wff to Conjunctive Normal Form.
Theorem	fald 39029	Refutation of falsity, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ¬ ⊥)
Theorem	tsim1 39030	A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 → 𝜓)))
Theorem	tsim2 39031	A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜑 ∨ (𝜑 → 𝜓)))
Theorem	tsim3 39032	A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (¬ 𝜓 ∨ (𝜑 → 𝜓)))
Theorem	tsbi1 39033	A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓)))
Theorem	tsbi2 39034	A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓)))
Theorem	tsbi3 39035	A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))
Theorem	tsbi4 39036	A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))
Theorem	tsxo1 39037	A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊻ 𝜓)))
Theorem	tsxo2 39038	A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ⊻ 𝜓)))
Theorem	tsxo3 39039	A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ⊻ 𝜓)))
Theorem	tsxo4 39040	A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ (𝜑 ⊻ 𝜓)))
Theorem	tsan1 39041	A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
Theorem	tsan2 39042	A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜑 ∨ ¬ (𝜑 ∧ 𝜓)))
Theorem	tsan3 39043	A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜓 ∨ ¬ (𝜑 ∧ 𝜓)))
Theorem	tsna1 39044	A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓)))
Theorem	tsna2 39045	A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜑 ∨ (𝜑 ⊼ 𝜓)))
Theorem	tsna3 39046	A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜓 ∨ (𝜑 ⊼ 𝜓)))
Theorem	tsor1 39047	A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
Theorem	tsor2 39048	A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ 𝜑 ∨ (𝜑 ∨ 𝜓)))
Theorem	tsor3 39049	A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ 𝜓 ∨ (𝜑 ∨ 𝜓)))
Theorem	ts3an1 39050	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 39051	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 39052	A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (𝜒 ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)))
Theorem	ts3or1 39053	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 39054	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 39055	A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ 𝜒 ∨ (𝜑 ∨ 𝜓 ∨ 𝜒)))
21.26.3  Equality deductions A collection of theorems for commuting equalities (or biconditionals) with other constructs.
Theorem	iuneq2f 39056	Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
Theorem	rabeq12f 39057	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 39058	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 39059	Equality deduction for substitution. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ ((𝑥 = 𝑦 ∧ ∀𝑧(𝜑 ↔ 𝜓)) → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜓))
Theorem	ralbi12f 39060	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 39061	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 39062*	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 39063	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 39064	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 39065	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 39066	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))
21.26.4  Miscellanea Work in progress or things that do not belong anywhere else.
Theorem	orcomdd 39067	Commutativity of logic disjunction, in double deduction form. Should not be moved to main, see PR #3034 in Github. Use orcomd 885 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 39068*	A version of scottex 9914 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 39069*	A version of scott0b 9918 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 39070*	A version of scott0f 39069 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 39071*	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 39072*	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 39073*	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)
21.27  Mathbox for Peter Mazsa
21.27.1  Notations
Syntax	cxrn 39074	Extend the definition of a class to include the range Cartesian product class.
class (𝐴 ⋉ 𝐵)
Syntax	cqmap 39075	Extend the definition of a class to include the quotient map of a class.
class QMap 𝑅
Syntax	cadjliftmap 39076	Extend the definition of a class to include the class of adjoined lift maps.
class (𝑅 AdjLiftMap 𝐴)
Syntax	cblockliftmap 39077	Extend the definition of a class to include the class of block lift maps.
class (𝑅 BlockLiftMap 𝐴)
Syntax	csucmap 39078	Extend the definition of a class to include the class of successor maps.
class SucMap
Syntax	csuccl 39079	Extend the definition of a class to include the class of successors.
class Suc
Syntax	cpre 39080	Extend the definition of a class to include the predecessor of a class.
class pre 𝑁
Syntax	cblockliftfix 39081	Extend the definition of a class to include the class of equilibrium block lifts.
class BlockLiftFix
Syntax	cshiftstable 39082	Extend the definition of a class to include the shift stability class.
class (𝑆 ShiftStable 𝐹)
Syntax	ccoss 39083	Extend the definition of a class to include the class of cosets by a class. (Read: the class of cosets by 𝑅.)
class ≀ 𝑅
Syntax	ccoels 39084	Extend the definition of a class to include the class of coelements on a class. (Read: the class of coelements on 𝐴.)
class ∼ 𝐴
Syntax	crels 39085	Extend the definition of a class to include the relation class.
class Rels
Syntax	cssr 39086	Extend the definition of a class to include the subset class.
class S
Syntax	crefs 39087	Extend the definition of a class to include the reflexivity class.
class Refs
Syntax	crefrels 39088	Extend the definition of a class to include the reflexive relations class.
class RefRels
Syntax	wrefrel 39089	Extend the definition of a wff to include the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.)
wff RefRel 𝑅
Syntax	ccnvrefs 39090	Extend the definition of a class to include the converse reflexivity class.
class CnvRefs
Syntax	ccnvrefrels 39091	Extend the definition of a class to include the converse reflexive relations class.
class CnvRefRels
Syntax	wcnvrefrel 39092	Extend the definition of a wff to include the converse reflexive relation predicate. (Read: 𝑅 is a converse reflexive relation.)
wff CnvRefRel 𝑅
Syntax	csyms 39093	Extend the definition of a class to include the symmetry class.
class Syms
Syntax	csymrels 39094	Extend the definition of a class to include the symmetry relations class.
class SymRels
Syntax	wsymrel 39095	Extend the definition of a wff to include the symmetry relation predicate. (Read: 𝑅 is a symmetric relation.)
wff SymRel 𝑅
Syntax	ctrs 39096	Extend the definition of a class to include the transitivity class (but cf. the transitive class defined in df-tr 5213).
class Trs
Syntax	ctrrels 39097	Extend the definition of a class to include the transitive relations class.
class TrRels
Syntax	wtrrel 39098	Extend the definition of a wff to include the transitive relation predicate. (Read: 𝑅 is a transitive relation.)
wff TrRel 𝑅
Syntax	ceqvrels 39099	Extend the definition of a class to include the equivalence relations class.
class EqvRels
Syntax	weqvrel 39100	Extend the definition of a wff to include the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.)
wff EqvRel 𝑅