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Theorem List for Metamath Proof Explorer - 37001-37100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtsbi1 37001 A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ ((Β¬ πœ‘ ∨ Β¬ πœ“) ∨ (πœ‘ ↔ πœ“)))
 
Theoremtsbi2 37002 A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ ((πœ‘ ∨ πœ“) ∨ (πœ‘ ↔ πœ“)))
 
Theoremtsbi3 37003 A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ ((πœ‘ ∨ Β¬ πœ“) ∨ Β¬ (πœ‘ ↔ πœ“)))
 
Theoremtsbi4 37004 A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ ((Β¬ πœ‘ ∨ πœ“) ∨ Β¬ (πœ‘ ↔ πœ“)))
 
Theoremtsxo1 37005 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ ((Β¬ πœ‘ ∨ Β¬ πœ“) ∨ Β¬ (πœ‘ ⊻ πœ“)))
 
Theoremtsxo2 37006 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ ((πœ‘ ∨ πœ“) ∨ Β¬ (πœ‘ ⊻ πœ“)))
 
Theoremtsxo3 37007 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ ((πœ‘ ∨ Β¬ πœ“) ∨ (πœ‘ ⊻ πœ“)))
 
Theoremtsxo4 37008 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ ((Β¬ πœ‘ ∨ πœ“) ∨ (πœ‘ ⊻ πœ“)))
 
Theoremtsan1 37009 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ ((Β¬ πœ‘ ∨ Β¬ πœ“) ∨ (πœ‘ ∧ πœ“)))
 
Theoremtsan2 37010 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ (πœ‘ ∨ Β¬ (πœ‘ ∧ πœ“)))
 
Theoremtsan3 37011 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ (πœ“ ∨ Β¬ (πœ‘ ∧ πœ“)))
 
Theoremtsna1 37012 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ ((Β¬ πœ‘ ∨ Β¬ πœ“) ∨ Β¬ (πœ‘ ⊼ πœ“)))
 
Theoremtsna2 37013 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ (πœ‘ ∨ (πœ‘ ⊼ πœ“)))
 
Theoremtsna3 37014 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ (πœ“ ∨ (πœ‘ ⊼ πœ“)))
 
Theoremtsor1 37015 A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(πœƒ β†’ ((πœ‘ ∨ πœ“) ∨ Β¬ (πœ‘ ∨ πœ“)))
 
Theoremtsor2 37016 A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(πœƒ β†’ (Β¬ πœ‘ ∨ (πœ‘ ∨ πœ“)))
 
Theoremtsor3 37017 A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(πœƒ β†’ (Β¬ πœ“ ∨ (πœ‘ ∨ πœ“)))
 
Theoremts3an1 37018 A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(πœƒ β†’ ((Β¬ (πœ‘ ∧ πœ“) ∨ Β¬ πœ’) ∨ (πœ‘ ∧ πœ“ ∧ πœ’)))
 
Theoremts3an2 37019 A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(πœƒ β†’ ((πœ‘ ∧ πœ“) ∨ Β¬ (πœ‘ ∧ πœ“ ∧ πœ’)))
 
Theoremts3an3 37020 A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(πœƒ β†’ (πœ’ ∨ Β¬ (πœ‘ ∧ πœ“ ∧ πœ’)))
 
Theoremts3or1 37021 A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(πœƒ β†’ (((πœ‘ ∨ πœ“) ∨ πœ’) ∨ Β¬ (πœ‘ ∨ πœ“ ∨ πœ’)))
 
Theoremts3or2 37022 A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(πœƒ β†’ (Β¬ (πœ‘ ∨ πœ“) ∨ (πœ‘ ∨ πœ“ ∨ πœ’)))
 
Theoremts3or3 37023 A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(πœƒ β†’ (Β¬ πœ’ ∨ (πœ‘ ∨ πœ“ ∨ πœ’)))
 
21.23.3  Equality deductions

A collection of theorems for commuting equalities (or biconditionals) with other constructs.

 
Theoremiuneq2f 37024 Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    β‡’   (𝐴 = 𝐡 β†’ βˆͺ π‘₯ ∈ 𝐴 𝐢 = βˆͺ π‘₯ ∈ 𝐡 𝐢)
 
Theoremrabeq12f 37025 Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    β‡’   ((𝐴 = 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 (πœ‘ ↔ πœ“)) β†’ {π‘₯ ∈ 𝐴 ∣ πœ‘} = {π‘₯ ∈ 𝐡 ∣ πœ“})
 
Theoremcsbeq12 37026 Equality deduction for substitution in class. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
((𝐴 = 𝐡 ∧ βˆ€π‘₯ 𝐢 = 𝐷) β†’ ⦋𝐴 / π‘₯⦌𝐢 = ⦋𝐡 / π‘₯⦌𝐷)
 
Theoremsbeqi 37027 Equality deduction for substitution. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
((π‘₯ = 𝑦 ∧ βˆ€π‘§(πœ‘ ↔ πœ“)) β†’ ([π‘₯ / 𝑧]πœ‘ ↔ [𝑦 / 𝑧]πœ“))
 
Theoremralbi12f 37028 Equality deduction for restricted universal quantification. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    β‡’   ((𝐴 = 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 (πœ‘ ↔ πœ“)) β†’ (βˆ€π‘₯ ∈ 𝐴 πœ‘ ↔ βˆ€π‘₯ ∈ 𝐡 πœ“))
 
Theoremoprabbi 37029 Equality deduction for class abstraction of nested ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
(βˆ€π‘₯βˆ€π‘¦βˆ€π‘§(πœ‘ ↔ πœ“) β†’ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ πœ‘} = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ πœ“})
 
Theoremmpobi123f 37030* Equality deduction for maps-to notations with two arguments. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    &   β„²π‘¦π΄    &   β„²π‘¦π΅    &   β„²π‘¦πΆ    &   β„²π‘¦π·    &   β„²π‘₯𝐢    &   β„²π‘₯𝐷    β‡’   (((𝐴 = 𝐡 ∧ 𝐢 = 𝐷) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐢 𝐸 = 𝐹) β†’ (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ 𝐸) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ 𝐹))
 
Theoremiuneq12f 37031 Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    β‡’   ((𝐴 = 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 𝐢 = 𝐷) β†’ βˆͺ π‘₯ ∈ 𝐴 𝐢 = βˆͺ π‘₯ ∈ 𝐡 𝐷)
 
Theoremiineq12f 37032 Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    β‡’   ((𝐴 = 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 𝐢 = 𝐷) β†’ ∩ π‘₯ ∈ 𝐴 𝐢 = ∩ π‘₯ ∈ 𝐡 𝐷)
 
Theoremopabbi 37033 Equality deduction for class abstraction of ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
(βˆ€π‘₯βˆ€π‘¦(πœ‘ ↔ πœ“) β†’ {⟨π‘₯, π‘¦βŸ© ∣ πœ‘} = {⟨π‘₯, π‘¦βŸ© ∣ πœ“})
 
Theoremmptbi12f 37034 Equality deduction for maps-to notations. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    β‡’   ((𝐴 = 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 𝐷 = 𝐸) β†’ (π‘₯ ∈ 𝐴 ↦ 𝐷) = (π‘₯ ∈ 𝐡 ↦ 𝐸))
 
21.23.4  Miscellanea

Work in progress or things that do not belong anywhere else.

 
Theoremorcomdd 37035 Commutativity of logic disjunction, in double deduction form. Should not be moved to main, see PR #3034 in Github. Use orcomd 870 instead. (Contributed by Giovanni Mascellani, 19-Mar-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(πœ‘ β†’ (πœ“ β†’ (πœ’ ∨ πœƒ)))    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœƒ ∨ πœ’)))
 
Theoremscottexf 37036* A version of scottex 9880 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
Ⅎ𝑦𝐴    &   β„²π‘₯𝐴    β‡’   {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} ∈ V
 
Theoremscott0f 37037* A version of scott0 9881 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
Ⅎ𝑦𝐴    &   β„²π‘₯𝐴    β‡’   (𝐴 = βˆ… ↔ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} = βˆ…)
 
Theoremscottn0f 37038* A version of scott0f 37037 with inequalities instead of equalities. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
Ⅎ𝑦𝐴    &   β„²π‘₯𝐴    β‡’   (𝐴 β‰  βˆ… ↔ {π‘₯ ∈ 𝐴 ∣ βˆ€π‘¦ ∈ 𝐴 (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘¦)} β‰  βˆ…)
 
Theoremac6s3f 37039* Generalization of the Axiom of Choice to classes, with bound-variable hypothesis. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
β„²π‘¦πœ“    &   π΄ ∈ V    &   (𝑦 = (π‘“β€˜π‘₯) β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦πœ‘ β†’ βˆƒπ‘“βˆ€π‘₯ ∈ 𝐴 πœ“)
 
Theoremac6s6 37040* Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
β„²π‘¦πœ“    &   π΄ ∈ V    &   (𝑦 = (π‘“β€˜π‘₯) β†’ (πœ‘ ↔ πœ“))    β‡’   βˆƒπ‘“βˆ€π‘₯ ∈ 𝐴 (βˆƒπ‘¦πœ‘ β†’ πœ“)
 
Theoremac6s6f 37041* Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 20-Aug-2018.)
𝐴 ∈ V    &   β„²π‘¦πœ“    &   (𝑦 = (π‘“β€˜π‘₯) β†’ (πœ‘ ↔ πœ“))    &   β„²π‘₯𝐴    β‡’   βˆƒπ‘“βˆ€π‘₯ ∈ 𝐴 (βˆƒπ‘¦πœ‘ β†’ πœ“)
 
21.24  Mathbox for Peter Mazsa
 
21.24.1  Notations
 
Syntaxcxrn 37042 Extend the definition of a class to include the range Cartesian product class.
class (𝐴 ⋉ 𝐡)
 
Syntaxccoss 37043 Extend the definition of a class to include the class of cosets by a class. (Read: the class of cosets by 𝑅.)
class ≀ 𝑅
 
Syntaxccoels 37044 Extend the definition of a class to include the class of coelements on a class. (Read: the class of coelements on 𝐴.)
class ∼ 𝐴
 
Syntaxcrels 37045 Extend the definition of a class to include the relation class.
class Rels
 
Syntaxcssr 37046 Extend the definition of a class to include the subset class.
class S
 
Syntaxcrefs 37047 Extend the definition of a class to include the reflexivity class.
class Refs
 
Syntaxcrefrels 37048 Extend the definition of a class to include the reflexive relations class.
class RefRels
 
Syntaxwrefrel 37049 Extend the definition of a wff to include the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.)
wff RefRel 𝑅
 
Syntaxccnvrefs 37050 Extend the definition of a class to include the converse reflexivity class.
class CnvRefs
 
Syntaxccnvrefrels 37051 Extend the definition of a class to include the converse reflexive relations class.
class CnvRefRels
 
Syntaxwcnvrefrel 37052 Extend the definition of a wff to include the converse reflexive relation predicate. (Read: 𝑅 is a converse reflexive relation.)
wff CnvRefRel 𝑅
 
Syntaxcsyms 37053 Extend the definition of a class to include the symmetry class.
class Syms
 
Syntaxcsymrels 37054 Extend the definition of a class to include the symmetry relations class.
class SymRels
 
Syntaxwsymrel 37055 Extend the definition of a wff to include the symmetry relation predicate. (Read: 𝑅 is a symmetric relation.)
wff SymRel 𝑅
 
Syntaxctrs 37056 Extend the definition of a class to include the transitivity class (but cf. the transitive class defined in df-tr 5267).
class Trs
 
Syntaxctrrels 37057 Extend the definition of a class to include the transitive relations class.
class TrRels
 
Syntaxwtrrel 37058 Extend the definition of a wff to include the transitive relation predicate. (Read: 𝑅 is a transitive relation.)
wff TrRel 𝑅
 
Syntaxceqvrels 37059 Extend the definition of a class to include the equivalence relations class.
class EqvRels
 
Syntaxweqvrel 37060 Extend the definition of a wff to include the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.)
wff EqvRel 𝑅
 
Syntaxccoeleqvrels 37061 Extend the definition of a class to include the coelement equivalence relations class.
class CoElEqvRels
 
Syntaxwcoeleqvrel 37062 Extend the definition of a wff to include the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.)
wff CoElEqvRel 𝐴
 
Syntaxcredunds 37063 Extend the definition of a class to include the redundancy class.
class Redunds
 
Syntaxwredund 37064 Extend the definition of a wff to include the redundancy predicate. (Read: 𝐴 is redundant with respect to 𝐡 in 𝐢.)
wff 𝐴 Redund ⟨𝐡, 𝐢⟩
 
Syntaxwredundp 37065 Extend wff definition to include the redundancy operator for propositions.
wff redund (πœ‘, πœ“, πœ’)
 
Syntaxcdmqss 37066 Extend the definition of a class to include the domain quotients class.
class DomainQss
 
Syntaxwdmqs 37067 Extend the definition of a wff to include the domain quotient predicate. (Read: the domain quotient of 𝑅 is 𝐴.)
wff 𝑅 DomainQs 𝐴
 
Syntaxcers 37068 Extend the definition of a class to include the equivalence relations on their domain quotients class.
class Ers
 
SyntaxwerALTV 37069 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
 
Syntaxccomembers 37070 Extend the definition of a class to include the comember equivalence relations class.
class CoMembErs
 
Syntaxwcomember 37071 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
 
Syntaxcfunss 37072 Extend the definition of a class to include the function set class.
class Funss
 
SyntaxcfunsALTV 37073 Extend the definition of a class to include the functions class, i.e., the function relations class.
class FunsALTV
 
SyntaxwfunALTV 37074 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
 
Syntaxcdisjss 37075 Extend the definition of a class to include the disjoint set class.
class Disjss
 
Syntaxcdisjs 37076 Extend the definition of a class to include the disjoints class, i.e., the disjoint relations class.
class Disjs
 
SyntaxwdisjALTV 37077 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
 
Syntaxceldisjs 37078 Extend the definition of a class to include the disjoint elements class, i.e., the disjoint element relations class.
class ElDisjs
 
Syntaxweldisj 37079 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
 
Syntaxwantisymrel 37080 Extend the definition of a wff to include the antisymmetry relation predicate. (Read: 𝑅 is an antisymmetric relation.)
wff AntisymRel 𝑅
 
Syntaxcparts 37081 Extend the definition of a class to include the partitions class, i.e., the partition relations class.
class Parts
 
Syntaxwpart 37082 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
 
Syntaxcmembparts 37083 Extend the definition of a class to include the member partitions class, i.e., the member partition relations class.
class MembParts
 
Syntaxwmembpart 37084 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
 
Theoremel2v1 37085 New way (elv 3481, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 23-Oct-2018.)
((π‘₯ ∈ V ∧ πœ‘) β†’ πœ“)    β‡’   (πœ‘ β†’ πœ“)
 
Theoremel3v 37086 New way (elv 3481, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. Inference forms (with 𝐴 ∈ V, 𝐡 ∈ V and 𝐢 ∈ V hypotheses) of the general theorems (proving ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š ∧ 𝐢 ∈ 𝑋) β†’ assertions) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018.)
((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ πœ‘)    β‡’   πœ‘
 
Theoremel3v1 37087 New way (elv 3481, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
((π‘₯ ∈ V ∧ πœ“ ∧ πœ’) β†’ πœƒ)    β‡’   ((πœ“ ∧ πœ’) β†’ πœƒ)
 
Theoremel3v2 37088 New way (elv 3481, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
((πœ‘ ∧ 𝑦 ∈ V ∧ πœ’) β†’ πœƒ)    β‡’   ((πœ‘ ∧ πœ’) β†’ πœƒ)
 
Theoremel3v3 37089 New way (elv 3481, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
((πœ‘ ∧ πœ“ ∧ 𝑧 ∈ V) β†’ πœƒ)    β‡’   ((πœ‘ ∧ πœ“) β†’ πœƒ)
 
Theoremel3v12 37090 New way (elv 3481, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
((π‘₯ ∈ V ∧ 𝑦 ∈ V ∧ πœ’) β†’ πœƒ)    β‡’   (πœ’ β†’ πœƒ)
 
Theoremel3v13 37091 New way (elv 3481, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
((π‘₯ ∈ V ∧ πœ“ ∧ 𝑧 ∈ V) β†’ πœƒ)    β‡’   (πœ“ β†’ πœƒ)
 
Theoremel3v23 37092 New way (elv 3481, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
((πœ‘ ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) β†’ πœƒ)    β‡’   (πœ‘ β†’ πœƒ)
 
Theoremanan 37093 Multiple commutations in conjunction. (Contributed by Peter Mazsa, 7-Mar-2020.)
((((πœ‘ ∧ πœ“) ∧ πœ’) ∧ ((πœ‘ ∧ πœƒ) ∧ 𝜏)) ↔ ((πœ“ ∧ πœƒ) ∧ (πœ‘ ∧ (πœ’ ∧ 𝜏))))
 
Theoremtriantru3 37094 A wff is equivalent to its conjunctions with truths. (Contributed by Peter Mazsa, 30-Nov-2018.)
πœ‘    &   πœ“    β‡’   (πœ’ ↔ (πœ‘ ∧ πœ“ ∧ πœ’))
 
Theorembianbi 37095 Exchanging conjunction in a biconditional. (Contributed by Peter Mazsa, 31-Jul-2023.)
(πœ‘ ↔ (πœ“ ∧ πœ’))    &   (πœ“ ↔ πœƒ)    β‡’   (πœ‘ ↔ (πœƒ ∧ πœ’))
 
Theorembianim 37096 Exchanging conjunction in a biconditional. (Contributed by Peter Mazsa, 31-Jul-2023.)
(πœ‘ ↔ (πœ“ ∧ πœ’))    &   (πœ’ β†’ (πœ“ ↔ πœƒ))    β‡’   (πœ‘ ↔ (πœƒ ∧ πœ’))
 
Theorembiorfd 37097 A wff is equivalent to its disjunction with falsehood, deduction form. (Contributed by Peter Mazsa, 22-Aug-2023.)
(πœ‘ β†’ Β¬ πœ“)    β‡’   (πœ‘ β†’ (πœ’ ↔ (πœ“ ∨ πœ’)))
 
Theoremeqbrtr 37098 Substitution of equal classes in binary relation. (Contributed by Peter Mazsa, 14-Jun-2024.)
((𝐴 = 𝐡 ∧ 𝐡𝑅𝐢) β†’ 𝐴𝑅𝐢)
 
Theoremeqbrb 37099 Substitution of equal classes in a binary relation. (Contributed by Peter Mazsa, 14-Jun-2024.)
((𝐴 = 𝐡 ∧ 𝐴𝑅𝐢) ↔ (𝐴 = 𝐡 ∧ 𝐡𝑅𝐢))
 
Theoremeqeltr 37100 Substitution of equal classes into element relation. (Contributed by Peter Mazsa, 22-Jul-2017.)
((𝐴 = 𝐡 ∧ 𝐡 ∈ 𝐢) β†’ 𝐴 ∈ 𝐢)
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