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Theorem List for Intuitionistic Logic Explorer - 3701-3800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssiun2s 3701* Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
(𝑥 = 𝐶𝐵 = 𝐷)       (𝐶𝐴𝐷 𝑥𝐴 𝐵)
 
Theoremiunss2 3702* A subclass condition on the members of two indexed classes 𝐶(𝑥) and 𝐷(𝑦) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 3611. (Contributed by NM, 9-Dec-2004.)
(∀𝑥𝐴𝑦𝐵 𝐶𝐷 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
 
Theoremiunab 3703* The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑}
 
Theoremiunrab 3704* The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑}
 
Theoremiunxdif2 3705* Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
(𝑥 = 𝑦𝐶 = 𝐷)       (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶)
 
Theoremssiinf 3706 Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
𝑥𝐶       (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
 
Theoremssiin 3707* Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
(𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
 
Theoremiinss 3708* Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
 
Theoremiinss2 3709 An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
(𝑥𝐴 𝑥𝐴 𝐵𝐵)
 
Theoremuniiun 3710* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
𝐴 = 𝑥𝐴 𝑥
 
Theoremintiin 3711* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
𝐴 = 𝑥𝐴 𝑥
 
Theoremiunid 3712* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
𝑥𝐴 {𝑥} = 𝐴
 
Theoremiun0 3713 An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑥𝐴 ∅ = ∅
 
Theorem0iun 3714 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑥 ∈ ∅ 𝐴 = ∅
 
Theorem0iin 3715 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
𝑥 ∈ ∅ 𝐴 = V
 
Theoremviin 3716* Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.)
𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦𝐴}
 
Theoremiunn0m 3717* There is an inhabited class in an indexed collection 𝐵(𝑥) iff the indexed union of them is inhabited. (Contributed by Jim Kingdon, 16-Aug-2018.)
(∃𝑥𝐴𝑦 𝑦𝐵 ↔ ∃𝑦 𝑦 𝑥𝐴 𝐵)
 
Theoremiinab 3718* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑}
 
Theoremiinrabm 3719* Indexed intersection of a restricted class builder. (Contributed by Jim Kingdon, 16-Aug-2018.)
(∃𝑥 𝑥𝐴 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑})
 
Theoremiunin2 3720* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3710 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
 
Theoremiunin1 3721* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3710 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵)
 
Theoremiundif2ss 3722* Indexed union of class difference. Compare to theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
𝑥𝐴 (𝐵𝐶) ⊆ (𝐵 𝑥𝐴 𝐶)
 
Theorem2iunin 3723* Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
𝑥𝐴 𝑦𝐵 (𝐶𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
 
Theoremiindif2m 3724* Indexed intersection of class difference. Compare to Theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
(∃𝑥 𝑥𝐴 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
 
Theoremiinin2m 3725* Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
(∃𝑥 𝑥𝐴 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
 
Theoremiinin1m 3726* Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
(∃𝑥 𝑥𝐴 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵))
 
Theoremelriin 3727* Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
 
Theoremriin0 3728* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)
 
Theoremriinm 3729* Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)
((∀𝑥𝑋 𝑆𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
 
Theoremiinxsng 3730* A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 
Theoremiinxprg 3731* Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
 
Theoremiunxsng 3732* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 
Theoremiunxsn 3733* A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)
𝐴 ∈ V    &   (𝑥 = 𝐴𝐵 = 𝐶)        𝑥 ∈ {𝐴}𝐵 = 𝐶
 
Theoremiunun 3734 Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
 
Theoremiunxun 3735 Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑥 ∈ (𝐴𝐵)𝐶 = ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
 
Theoremiunxiun 3736* Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
𝑥 𝑦𝐴 𝐵𝐶 = 𝑦𝐴 𝑥𝐵 𝐶
 
Theoremiinuniss 3737* A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33 but with equality changed to subset. (Contributed by Jim Kingdon, 19-Aug-2018.)
(𝐴 𝐵) ⊆ 𝑥𝐵 (𝐴𝑥)
 
Theoremiununir 3738* A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33 but with biconditional changed to implication. (Contributed by Jim Kingdon, 19-Aug-2018.)
((𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
 
Theoremsspwuni 3739 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
 
Theorempwssb 3740* Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
(𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 
Theoremelpwuni 3741 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(𝐵𝐴 → (𝐴 ⊆ 𝒫 𝐵 𝐴 = 𝐵))
 
Theoremiinpw 3742* The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥
 
Theoremiunpwss 3743* Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)
𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴
 
Theoremrintm 3744* Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥𝑋) → (𝐴 𝑋) = 𝑋)
 
2.1.21  Disjointness
 
Syntaxwdisj 3745 Extend wff notation to include the statement that a family of classes 𝐵(𝑥), for 𝑥𝐴, is a disjoint family.
wff Disj 𝑥𝐴 𝐵
 
Definitiondf-disj 3746* A collection of classes 𝐵(𝑥) is disjoint when for each element 𝑦, it is in 𝐵(𝑥) for at most one 𝑥. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
(Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
 
Theoremdfdisj2 3747* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
(Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
 
Theoremdisjss2 3748 If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
 
Theoremdisjeq2 3749 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
 
Theoremdisjeq2dv 3750* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
 
Theoremdisjss1 3751* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
 
Theoremdisjeq1 3752* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
 
Theoremdisjeq1d 3753* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
 
Theoremdisjeq12d 3754* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))
 
Theoremcbvdisj 3755* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
 
Theoremcbvdisjv 3756* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
(𝑥 = 𝑦𝐵 = 𝐶)       (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
 
Theoremnfdisjv 3757* Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
𝑦𝐴    &   𝑦𝐵       𝑦Disj 𝑥𝐴 𝐵
 
Theoremnfdisj1 3758 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑥Disj 𝑥𝐴 𝐵
 
Theoreminvdisj 3759* If there is a function 𝐶(𝑦) such that 𝐶(𝑦) = 𝑥 for all 𝑦𝐵(𝑥), then the sets 𝐵(𝑥) for distinct 𝑥𝐴 are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
(∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥Disj 𝑥𝐴 𝐵)
 
Theoremsndisj 3760 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥𝐴 {𝑥}
 
Theorem0disj 3761 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥𝐴
 
Theoremdisjxsn 3762* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥 ∈ {𝐴}𝐵
 
Theoremdisjx0 3763 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥 ∈ ∅ 𝐵
 
2.1.22  Binary relations
 
Syntaxwbr 3764 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 3765 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 4600). (Contributed by NM, 31-Dec-1993.)
(𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
 
Theorembreq 3766 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
(𝑅 = 𝑆 → (𝐴𝑅𝐵𝐴𝑆𝐵))
 
Theorembreq1 3767 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))
 
Theorembreq2 3768 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(𝐴 = 𝐵 → (𝐶𝑅𝐴𝐶𝑅𝐵))
 
Theorembreq12 3769 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreqi 3770 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
𝑅 = 𝑆       (𝐴𝑅𝐵𝐴𝑆𝐵)
 
Theorembreq1i 3771 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
𝐴 = 𝐵       (𝐴𝑅𝐶𝐵𝑅𝐶)
 
Theorembreq2i 3772 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
𝐴 = 𝐵       (𝐶𝑅𝐴𝐶𝑅𝐵)
 
Theorembreq12i 3773 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 3774 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐶))
 
Theorembreqd 3775 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
 
Theorembreq2d 3776 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝑅𝐴𝐶𝑅𝐵))
 
Theorembreq12d 3777 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 3778 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝑅 = 𝑆)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝑅𝐶𝐵𝑆𝐷))
 
Theorembreqan12d 3779 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreqan12rd 3780 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theoremnbrne1 3781 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 3782 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 3783 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐵𝑅𝐶       𝐴𝑅𝐶
 
Theoremeqbrtrd 3784 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrri 3785 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐴𝑅𝐶       𝐵𝑅𝐶
 
Theoremeqbrtrrd 3786 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝑅𝐶)       (𝜑𝐵𝑅𝐶)
 
Theorembreqtri 3787 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝑅𝐵    &   𝐵 = 𝐶       𝐴𝑅𝐶
 
Theorembreqtrd 3788 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrri 3789 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝑅𝐵    &   𝐶 = 𝐵       𝐴𝑅𝐶
 
Theorembreqtrrd 3790 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝑅𝐶)
 
Theorem3brtr3i 3791 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
𝐴𝑅𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝑅𝐷
 
Theorem3brtr4i 3792 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
𝐴𝑅𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝑅𝐷
 
Theorem3brtr3d 3793 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝑅𝐷)
 
Theorem3brtr4d 3794 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝑅𝐷)
 
Theorem3brtr3g 3795 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(𝜑𝐴𝑅𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝑅𝐷)
 
Theorem3brtr4g 3796 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(𝜑𝐴𝑅𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝑅𝐷)
 
Theoremsyl5eqbr 3797 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
𝐴 = 𝐵    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theoremsyl5eqbrr 3798 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
𝐵 = 𝐴    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theoremsyl5breq 3799 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
𝐴𝑅𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝑅𝐶)
 
Theoremsyl5breqr 3800 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
𝐴𝑅𝐵    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝑅𝐶)
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