P. 374 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37301-37400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tsim1 37301	A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 → 𝜓)))
Theorem	tsim2 37302	A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜑 ∨ (𝜑 → 𝜓)))
Theorem	tsim3 37303	A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (¬ 𝜓 ∨ (𝜑 → 𝜓)))
Theorem	tsbi1 37304	A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓)))
Theorem	tsbi2 37305	A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ (𝜑 ↔ 𝜓)))
Theorem	tsbi3 37306	A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))
Theorem	tsbi4 37307	A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)))
Theorem	tsxo1 37308	A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊻ 𝜓)))
Theorem	tsxo2 37309	A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ⊻ 𝜓)))
Theorem	tsxo3 37310	A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ⊻ 𝜓)))
Theorem	tsxo4 37311	A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ (𝜑 ⊻ 𝜓)))
Theorem	tsan1 37312	A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
Theorem	tsan2 37313	A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜑 ∨ ¬ (𝜑 ∧ 𝜓)))
Theorem	tsan3 37314	A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜓 ∨ ¬ (𝜑 ∧ 𝜓)))
Theorem	tsna1 37315	A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓)))
Theorem	tsna2 37316	A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜑 ∨ (𝜑 ⊼ 𝜓)))
Theorem	tsna3 37317	A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
⊢ (𝜃 → (𝜓 ∨ (𝜑 ⊼ 𝜓)))
Theorem	tsor1 37318	A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
Theorem	tsor2 37319	A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ 𝜑 ∨ (𝜑 ∨ 𝜓)))
Theorem	tsor3 37320	A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ 𝜓 ∨ (𝜑 ∨ 𝜓)))
Theorem	ts3an1 37321	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 37322	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 37323	A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (𝜒 ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)))
Theorem	ts3or1 37324	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 37325	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 37326	A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ 𝜒 ∨ (𝜑 ∨ 𝜓 ∨ 𝜒)))
21.23.3  Equality deductions A collection of theorems for commuting equalities (or biconditionals) with other constructs.
Theorem	iuneq2f 37327	Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
Theorem	rabeq12f 37328	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 37329	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 37330	Equality deduction for substitution. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ ((𝑥 = 𝑦 ∧ ∀𝑧(𝜑 ↔ 𝜓)) → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜓))
Theorem	ralbi12f 37331	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 37332	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 37333*	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 37334	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 37335	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 37336	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 37337	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))
21.23.4  Miscellanea Work in progress or things that do not belong anywhere else.
Theorem	orcomdd 37338	Commutativity of logic disjunction, in double deduction form. Should not be moved to main, see PR #3034 in Github. Use orcomd 867 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 37339*	A version of scottex 9882 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 37340*	A version of scott0 9883 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 37341*	A version of scott0f 37340 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 37342*	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 37343*	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 37344*	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)
21.24  Mathbox for Peter Mazsa
21.24.1  Notations
Syntax	cxrn 37345	Extend the definition of a class to include the range Cartesian product class.
class (𝐴 ⋉ 𝐵)
Syntax	ccoss 37346	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 37347	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 37348	Extend the definition of a class to include the relation class.
class Rels
Syntax	cssr 37349	Extend the definition of a class to include the subset class.
class S
Syntax	crefs 37350	Extend the definition of a class to include the reflexivity class.
class Refs
Syntax	crefrels 37351	Extend the definition of a class to include the reflexive relations class.
class RefRels
Syntax	wrefrel 37352	Extend the definition of a wff to include the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.)
wff RefRel 𝑅
Syntax	ccnvrefs 37353	Extend the definition of a class to include the converse reflexivity class.
class CnvRefs
Syntax	ccnvrefrels 37354	Extend the definition of a class to include the converse reflexive relations class.
class CnvRefRels
Syntax	wcnvrefrel 37355	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 37356	Extend the definition of a class to include the symmetry class.
class Syms
Syntax	csymrels 37357	Extend the definition of a class to include the symmetry relations class.
class SymRels
Syntax	wsymrel 37358	Extend the definition of a wff to include the symmetry relation predicate. (Read: 𝑅 is a symmetric relation.)
wff SymRel 𝑅
Syntax	ctrs 37359	Extend the definition of a class to include the transitivity class (but cf. the transitive class defined in df-tr 5265).
class Trs
Syntax	ctrrels 37360	Extend the definition of a class to include the transitive relations class.
class TrRels
Syntax	wtrrel 37361	Extend the definition of a wff to include the transitive relation predicate. (Read: 𝑅 is a transitive relation.)
wff TrRel 𝑅
Syntax	ceqvrels 37362	Extend the definition of a class to include the equivalence relations class.
class EqvRels
Syntax	weqvrel 37363	Extend the definition of a wff to include the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.)
wff EqvRel 𝑅
Syntax	ccoeleqvrels 37364	Extend the definition of a class to include the coelement equivalence relations class.
class CoElEqvRels
Syntax	wcoeleqvrel 37365	Extend the definition of a wff to include the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.)
wff CoElEqvRel 𝐴
Syntax	credunds 37366	Extend the definition of a class to include the redundancy class.
class Redunds
Syntax	wredund 37367	Extend the definition of a wff to include the redundancy predicate. (Read: 𝐴 is redundant with respect to 𝐵 in 𝐶.)
wff 𝐴 Redund ⟨𝐵, 𝐶⟩
Syntax	wredundp 37368	Extend wff definition to include the redundancy operator for propositions.
wff redund (𝜑, 𝜓, 𝜒)
Syntax	cdmqss 37369	Extend the definition of a class to include the domain quotients class.
class DomainQss
Syntax	wdmqs 37370	Extend the definition of a wff to include the domain quotient predicate. (Read: the domain quotient of 𝑅 is 𝐴.)
wff 𝑅 DomainQs 𝐴
Syntax	cers 37371	Extend the definition of a class to include the equivalence relations on their domain quotients class.
class Ers
Syntax	werALTV 37372	Extend the definition of a wff to include the equivalence relation on its domain quotient predicate. (Read: 𝑅 is an equivalence relation on its domain quotient 𝐴.)
wff 𝑅 ErALTV 𝐴
Syntax	ccomembers 37373	Extend the definition of a class to include the comember equivalence relations class.
class CoMembErs
Syntax	wcomember 37374	Extend the definition of a wff to include the comember equivalence relation predicate. (Read: the comember equivalence relation on 𝐴, or, the restricted coelement equivalence relation on its domain quotient 𝐴.)
wff CoMembEr 𝐴
Syntax	cfunss 37375	Extend the definition of a class to include the function set class.
class Funss
Syntax	cfunsALTV 37376	Extend the definition of a class to include the functions class, i.e., the function relations class.
class FunsALTV
Syntax	wfunALTV 37377	Extend the definition of a wff to include the function predicate, i.e., the function relation predicate. (Read: 𝐹 is a function.)
wff FunALTV 𝐹
Syntax	cdisjss 37378	Extend the definition of a class to include the disjoint set class.
class Disjss
Syntax	cdisjs 37379	Extend the definition of a class to include the disjoints class, i.e., the disjoint relations class.
class Disjs
Syntax	wdisjALTV 37380	Extend the definition of a wff to include the disjoint predicate, i.e., the disjoint relation predicate. (Read: 𝑅 is a disjoint.)
wff Disj 𝑅
Syntax	celdisjs 37381	Extend the definition of a class to include the disjoint elements class, i.e., the disjoint element relations class.
class ElDisjs
Syntax	weldisj 37382	Extend the definition of a wff to include the disjoint element predicate, i.e., the disjoint element relation predicate. (Read: the elements of 𝐴 are disjoint.)
wff ElDisj 𝐴
Syntax	wantisymrel 37383	Extend the definition of a wff to include the antisymmetry relation predicate. (Read: 𝑅 is an antisymmetric relation.)
wff AntisymRel 𝑅
Syntax	cparts 37384	Extend the definition of a class to include the partitions class, i.e., the partition relations class.
class Parts
Syntax	wpart 37385	Extend the definition of a wff to include the partition predicate, i.e., the partition relation predicate. (Read: 𝐴 is a partition by 𝑅.)
wff 𝑅 Part 𝐴
Syntax	cmembparts 37386	Extend the definition of a class to include the member partitions class, i.e., the member partition relations class.
class MembParts
Syntax	wmembpart 37387	Extend the definition of a wff to include the member partition predicate, i.e., the member partition relation predicate. (Read: 𝐴 is a member partition.)
wff MembPart 𝐴
21.24.2  Preparatory theorems
Theorem	el2v1 37388	New way (elv 3478, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 23-Oct-2018.)
⊢ ((𝑥 ∈ V ∧ 𝜑) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	el3v 37389	New way (elv 3478, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. Inference forms (with ⊢ 𝐴 ∈ V, ⊢ 𝐵 ∈ V and ⊢ 𝐶 ∈ V hypotheses) of the general theorems (proving ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → assertions) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018.)
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜑)    ⇒   ⊢ 𝜑
Theorem	el3v1 37390	New way (elv 3478, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
⊢ ((𝑥 ∈ V ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
Theorem	el3v2 37391	New way (elv 3478, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
⊢ ((𝜑 ∧ 𝑦 ∈ V ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
Theorem	el3v3 37392	New way (elv 3478, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ V) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	el3v12 37393	New way (elv 3478, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	el3v13 37394	New way (elv 3478, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ ((𝑥 ∈ V ∧ 𝜓 ∧ 𝑧 ∈ V) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)
Theorem	el3v23 37395	New way (elv 3478, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ ((𝜑 ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	anan 37396	Multiple commutations in conjunction. (Contributed by Peter Mazsa, 7-Mar-2020.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜑 ∧ 𝜃) ∧ 𝜏)) ↔ ((𝜓 ∧ 𝜃) ∧ (𝜑 ∧ (𝜒 ∧ 𝜏))))
Theorem	triantru3 37397	A wff is equivalent to its conjunctions with truths. (Contributed by Peter Mazsa, 30-Nov-2018.)
⊢ 𝜑    &   ⊢ 𝜓    ⇒   ⊢ (𝜒 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	bianbi 37398	Exchanging conjunction in a biconditional. (Contributed by Peter Mazsa, 31-Jul-2023.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (𝜑 ↔ (𝜃 ∧ 𝜒))
Theorem	bianim 37399	Exchanging conjunction in a biconditional. (Contributed by Peter Mazsa, 31-Jul-2023.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 ↔ (𝜃 ∧ 𝜒))
Theorem	biorfd 37400	A wff is equivalent to its disjunction with falsehood, deduction form. (Contributed by Peter Mazsa, 22-Aug-2023.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → (𝜒 ↔ (𝜓 ∨ 𝜒)))