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Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem0disj 4001 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥𝐴
 
Theoremdisjxsn 4002* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥 ∈ {𝐴}𝐵
 
Theoremdisjx0 4003 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥 ∈ ∅ 𝐵
 
2.1.22  Binary relations
 
Syntaxwbr 4004 Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous.
wff 𝐴𝑅𝐵
 
Definitiondf-br 4005 Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. This definition of relations is well-defined, although not very meaningful, when classes 𝐴 and/or 𝐵 are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when 𝑅 is a proper class (see for example iprc 4896). (Contributed by NM, 31-Dec-1993.)
(𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
 
Theorembreq 4006 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
(𝑅 = 𝑆 → (𝐴𝑅𝐵𝐴𝑆𝐵))
 
Theorembreq1 4007 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))
 
Theorembreq2 4008 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(𝐴 = 𝐵 → (𝐶𝑅𝐴𝐶𝑅𝐵))
 
Theorembreq12 4009 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreqi 4010 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
𝑅 = 𝑆       (𝐴𝑅𝐵𝐴𝑆𝐵)
 
Theorembreq1i 4011 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
𝐴 = 𝐵       (𝐴𝑅𝐶𝐵𝑅𝐶)
 
Theorembreq2i 4012 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
𝐴 = 𝐵       (𝐶𝑅𝐴𝐶𝑅𝐵)
 
Theorembreq12i 4013 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝑅𝐶𝐵𝑅𝐷)
 
Theorembreq1d 4014 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐶))
 
Theorembreqd 4015 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
 
Theorembreq2d 4016 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝑅𝐴𝐶𝑅𝐵))
 
Theorembreq12d 4017 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreq123d 4018 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝑅 = 𝑆)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝑅𝐶𝐵𝑆𝐷))
 
Theorembreqdi 4019 Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶𝐴𝐷)       (𝜑𝐶𝐵𝐷)
 
Theorembreqan12d 4020 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreqan12rd 4021 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theoremeqnbrtrd 4022 Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → ¬ 𝐵𝑅𝐶)       (𝜑 → ¬ 𝐴𝑅𝐶)
 
Theoremnbrne1 4023 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵𝐶)
 
Theoremnbrne2 4024 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
((𝐴𝑅𝐶 ∧ ¬ 𝐵𝑅𝐶) → 𝐴𝐵)
 
Theoremeqbrtri 4025 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐵𝑅𝐶       𝐴𝑅𝐶
 
Theoremeqbrtrd 4026 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrri 4027 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐴𝑅𝐶       𝐵𝑅𝐶
 
Theoremeqbrtrrd 4028 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝑅𝐶)       (𝜑𝐵𝑅𝐶)
 
Theorembreqtri 4029 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝑅𝐵    &   𝐵 = 𝐶       𝐴𝑅𝐶
 
Theorembreqtrd 4030 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrri 4031 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝑅𝐵    &   𝐶 = 𝐵       𝐴𝑅𝐶
 
Theorembreqtrrd 4032 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝑅𝐶)
 
Theorem3brtr3i 4033 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
𝐴𝑅𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝑅𝐷
 
Theorem3brtr4i 4034 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
𝐴𝑅𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝑅𝐷
 
Theorem3brtr3d 4035 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝑅𝐷)
 
Theorem3brtr4d 4036 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝑅𝐷)
 
Theorem3brtr3g 4037 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(𝜑𝐴𝑅𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝑅𝐷)
 
Theorem3brtr4g 4038 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(𝜑𝐴𝑅𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝑅𝐷)
 
Theoremeqbrtrid 4039 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
𝐴 = 𝐵    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrrid 4040 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
𝐵 = 𝐴    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrid 4041 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
𝐴𝑅𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrrid 4042 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
𝐴𝑅𝐵    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrdi 4043 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   𝐵𝑅𝐶       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrrdi 4044 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐵 = 𝐴)    &   𝐵𝑅𝐶       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrdi 4045 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrrdi 4046 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
(𝜑𝐴𝑅𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝑅𝐶)
 
Theoremssbrd 4047 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
 
Theoremssbri 4048 Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
𝐴𝐵       (𝐶𝐴𝐷𝐶𝐵𝐷)
 
Theoremnfbrd 4049 Deduction version of bound-variable hypothesis builder nfbr 4050. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝑅)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)
 
Theoremnfbr 4050 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝐴    &   𝑥𝑅    &   𝑥𝐵       𝑥 𝐴𝑅𝐵
 
Theorembrab1 4051* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
(𝑥𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
 
Theorembr0 4052 The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
¬ 𝐴𝐵
 
Theorembrne0 4053 If two sets are in a binary relation, the relation cannot be empty. In fact, the relation is also inhabited, as seen at brm 4054. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
(𝐴𝑅𝐵𝑅 ≠ ∅)
 
Theorembrm 4054* If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)
(𝐴𝑅𝐵 → ∃𝑥 𝑥𝑅)
 
Theorembrun 4055 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))
 
Theorembrin 4056 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))
 
Theorembrdif 4057 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))
 
Theoremsbcbrg 4058 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
 
Theoremsbcbr12g 4059* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐴 / 𝑥𝐶))
 
Theoremsbcbr1g 4060* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐶))
 
Theoremsbcbr2g 4061* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝑅𝐴 / 𝑥𝐶))
 
Theorembrralrspcev 4062* Restricted existential specialization with a restricted universal quantifier over a relation, closed form. (Contributed by AV, 20-Aug-2022.)
((𝐵𝑋 ∧ ∀𝑦𝑌 𝐴𝑅𝐵) → ∃𝑥𝑋𝑦𝑌 𝐴𝑅𝑥)
 
Theorembrimralrspcev 4063* Restricted existential specialization with a restricted universal quantifier over an implication with a relation in the antecedent, closed form. (Contributed by AV, 20-Aug-2022.)
((𝐵𝑋 ∧ ∀𝑦𝑌 ((𝜑𝐴𝑅𝐵) → 𝜓)) → ∃𝑥𝑋𝑦𝑌 ((𝜑𝐴𝑅𝑥) → 𝜓))
 
2.1.23  Ordered-pair class abstractions (class builders)
 
Syntaxcopab 4064 Extend class notation to include ordered-pair class abstraction (class builder).
class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Syntaxcmpt 4065 Extend the definition of a class to include maps-to notation for defining a function via a rule.
class (𝑥𝐴𝐵)
 
Definitiondf-opab 4066* Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually 𝑥 and 𝑦 are distinct, although the definition doesn't strictly require it. The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. (Contributed by NM, 4-Jul-1994.)
{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
 
Definitiondf-mpt 4067* Define maps-to notation for defining a function via a rule. Read as "the function defined by the map from 𝑥 (in 𝐴) to 𝐵(𝑥)". The class expression 𝐵 is the value of the function at 𝑥 and normally contains the variable 𝑥. Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.)
(𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
 
Theoremopabss 4068* The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
 
Theoremopabbid 4069 Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
 
Theoremopabbidv 4070* Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
 
Theoremopabbii 4071 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
(𝜑𝜓)       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
 
Theoremnfopab 4072* Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.)
𝑧𝜑       𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Theoremnfopab1 4073 The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Theoremnfopab2 4074 The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Theoremcbvopab 4075* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
 
Theoremcbvopabv 4076* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
 
Theoremcbvopab1 4077* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑧𝜑    &   𝑥𝜓    &   (𝑥 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
 
Theoremcbvopab2 4078* Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
𝑧𝜑    &   𝑦𝜓    &   (𝑦 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}
 
Theoremcbvopab1s 4079* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)
{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ [𝑧 / 𝑥]𝜑}
 
Theoremcbvopab1v 4080* Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
(𝑥 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
 
Theoremcbvopab2v 4081* Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
(𝑦 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}
 
Theoremcsbopabg 4082* Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
(𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
 
Theoremunopab 4083 Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)}
 
Theoremmpteq12f 4084 An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
Theoremmpteq12dva 4085* An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑𝐴 = 𝐶)    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
Theoremmpteq12dv 4086* An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)
(𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
Theoremmpteq12 4087* An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.)
((𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
Theoremmpteq1 4088* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
(𝐴 = 𝐵 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
 
Theoremmpteq1d 4089* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
 
Theoremmpteq2ia 4090 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
(𝑥𝐴𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑥𝐴𝐶)
 
Theoremmpteq2i 4091 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
𝐵 = 𝐶       (𝑥𝐴𝐵) = (𝑥𝐴𝐶)
 
Theoremmpteq12i 4092 An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)
𝐴 = 𝐶    &   𝐵 = 𝐷       (𝑥𝐴𝐵) = (𝑥𝐶𝐷)
 
Theoremmpteq2da 4093 Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
 
Theoremmpteq2dva 4094* Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
 
Theoremmpteq2dv 4095* An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
 
Theoremnfmpt 4096* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝑦𝐴𝐵)
 
Theoremnfmpt1 4097 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
𝑥(𝑥𝐴𝐵)
 
Theoremcbvmptf 4098* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Thierry Arnoux, 9-Mar-2017.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
 
Theoremcbvmpt 4099* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
 
Theoremcbvmptv 4100* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)
(𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
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