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Theorem List for Intuitionistic Logic Explorer - 4801-4900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnvcnv2 4801 The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
𝐴 = (𝐴 ↾ V)
 
Theoremcnvcnvss 4802 The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)
𝐴𝐴
 
Theoremcnveqb 4803 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))
 
Theoremcnveq0 4804 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
(Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅))
 
Theoremdfrel3 4805 Alternate definition of relation. (Contributed by NM, 14-May-2008.)
(Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)
 
Theoremdmresv 4806 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
dom (𝐴 ↾ V) = dom 𝐴
 
Theoremrnresv 4807 The range of a universal restriction. (Contributed by NM, 14-May-2008.)
ran (𝐴 ↾ V) = ran 𝐴
 
Theoremdfrn4 4808 Range defined in terms of image. (Contributed by NM, 14-May-2008.)
ran 𝐴 = (𝐴 “ V)
 
Theoremcsbrng 4809 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)
 
Theoremrescnvcnv 4810 The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵) = (𝐴𝐵)
 
Theoremcnvcnvres 4811 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
(𝐴𝐵) = (𝐴𝐵)
 
Theoremimacnvcnv 4812 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
(𝐴𝐵) = (𝐴𝐵)
 
Theoremdmsnm 4813* The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
(𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
 
Theoremrnsnm 4814* The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
(𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ ran {𝐴})
 
Theoremdmsn0 4815 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
dom {∅} = ∅
 
Theoremcnvsn0 4816 The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
{∅} = ∅
 
Theoremdmsn0el 4817 The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)
(∅ ∈ 𝐴 → dom {𝐴} = ∅)
 
Theoremrelsn2m 4818* A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.)
𝐴 ∈ V       (Rel {𝐴} ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
 
Theoremdmsnopg 4819 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)
(𝐵𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
 
Theoremdmpropg 4820 The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
((𝐵𝑉𝐷𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})
 
Theoremdmsnop 4821 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐵 ∈ V       dom {⟨𝐴, 𝐵⟩} = {𝐴}
 
Theoremdmprop 4822 The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)
𝐵 ∈ V    &   𝐷 ∈ V       dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}
 
Theoremdmtpop 4823 The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)
𝐵 ∈ V    &   𝐷 ∈ V    &   𝐹 ∈ V       dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = {𝐴, 𝐶, 𝐸}
 
Theoremcnvcnvsn 4824 Double converse of a singleton of an ordered pair. (Unlike cnvsn 4830, this does not need any sethood assumptions on 𝐴 and 𝐵.) (Contributed by Mario Carneiro, 26-Apr-2015.)
{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
 
Theoremdmsnsnsng 4825 The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.)
(𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
 
Theoremrnsnopg 4826 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝐴𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
 
Theoremrnpropg 4827 The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)
((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})
 
Theoremrnsnop 4828 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V       ran {⟨𝐴, 𝐵⟩} = {𝐵}
 
Theoremop1sta 4829 Extract the first member of an ordered pair. (See op2nda 4832 to extract the second member and op1stb 4236 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        dom {⟨𝐴, 𝐵⟩} = 𝐴
 
Theoremcnvsn 4830 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
 
Theoremop2ndb 4831 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4236 to extract the first member and op2nda 4832 for an alternate version.) (Contributed by NM, 25-Nov-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        {⟨𝐴, 𝐵⟩} = 𝐵
 
Theoremop2nda 4832 Extract the second member of an ordered pair. (See op1sta 4829 to extract the first member and op2ndb 4831 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝐴 ∈ V    &   𝐵 ∈ V        ran {⟨𝐴, 𝐵⟩} = 𝐵
 
Theoremcnvsng 4833 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
 
Theoremopswapg 4834 Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
 
Theoremelxp4 4835 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4836. (Contributed by NM, 17-Feb-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
 
Theoremelxp5 4836 Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4835 when the double intersection does not create class existence problems (caused by int0 3656). (Contributed by NM, 1-Aug-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))
 
Theoremcnvresima 4837 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
((𝐹𝐴) “ 𝐵) = ((𝐹𝐵) ∩ 𝐴)
 
Theoremresdm2 4838 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
(𝐴 ↾ dom 𝐴) = 𝐴
 
Theoremresdmres 4839 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
(𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
 
Theoremimadmres 4840 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
(𝐴 “ dom (𝐴𝐵)) = (𝐴𝐵)
 
Theoremmptpreima 4841* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐹 = (𝑥𝐴𝐵)       (𝐹𝐶) = {𝑥𝐴𝐵𝐶}
 
Theoremmptiniseg 4842* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐹 = (𝑥𝐴𝐵)       (𝐶𝑉 → (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 = 𝐶})
 
Theoremdmmpt 4843 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)
𝐹 = (𝑥𝐴𝐵)       dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
 
Theoremdmmptss 4844* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
𝐹 = (𝑥𝐴𝐵)       dom 𝐹𝐴
 
Theoremdmmptg 4845* The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)
(∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)
 
Theoremrelco 4846 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
Rel (𝐴𝐵)
 
Theoremdfco2 4847* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
(𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
 
Theoremdfco2a 4848* Generalization of dfco2 4847, where 𝐶 can have any value between dom 𝐴 ∩ ran 𝐵 and V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴𝐵) = 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
 
Theoremcoundi 4849 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴 ∘ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 
Theoremcoundir 4850 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴𝐵) ∘ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 
Theoremcores 4851 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(ran 𝐵𝐶 → ((𝐴𝐶) ∘ 𝐵) = (𝐴𝐵))
 
Theoremresco 4852 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
((𝐴𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵𝐶))
 
Theoremimaco 4853 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
((𝐴𝐵) “ 𝐶) = (𝐴 “ (𝐵𝐶))
 
Theoremrnco 4854 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
 
Theoremrnco2 4855 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
ran (𝐴𝐵) = (𝐴 “ ran 𝐵)
 
Theoremdmco 4856 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
dom (𝐴𝐵) = (𝐵 “ dom 𝐴)
 
Theoremcoiun 4857* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
(𝐴 𝑥𝐶 𝐵) = 𝑥𝐶 (𝐴𝐵)
 
Theoremcocnvcnv1 4858 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)
 
Theoremcocnvcnv2 4859 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)
 
Theoremcores2 4860 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
(dom 𝐴𝐶 → (𝐴(𝐵𝐶)) = (𝐴𝐵))
 
Theoremco02 4861 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
(𝐴 ∘ ∅) = ∅
 
Theoremco01 4862 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
(∅ ∘ 𝐴) = ∅
 
Theoremcoi1 4863 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
(Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
 
Theoremcoi2 4864 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
(Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)
 
Theoremcoires1 4865 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)
(𝐴 ∘ ( I ↾ 𝐵)) = (𝐴𝐵)
 
Theoremcoass 4866 Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)
((𝐴𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵𝐶))
 
Theoremrelcnvtr 4867 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
(Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
 
Theoremrelssdmrn 4868 A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
(Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
 
Theoremcnvssrndm 4869 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐴 ⊆ (ran 𝐴 × dom 𝐴)
 
Theoremcossxp 4870 Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)
 
Theoremrelrelss 4871 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))
 
Theoremunielrel 4872 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
((Rel 𝑅𝐴𝑅) → 𝐴 𝑅)
 
Theoremrelfld 4873 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
(Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
 
Theoremrelresfld 4874 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
(Rel 𝑅 → (𝑅 𝑅) = 𝑅)
 
Theoremrelcoi2 4875 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
(Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)
 
Theoremrelcoi1 4876 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)
(Rel 𝑅 → (𝑅 ∘ ( I ↾ 𝑅)) = 𝑅)
 
Theoremunidmrn 4877 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
𝐴 = (dom 𝐴 ∪ ran 𝐴)
 
Theoremrelcnvfld 4878 if 𝑅 is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
(Rel 𝑅 𝑅 = 𝑅)
 
Theoremdfdm2 4879 Alternate definition of domain df-dm 4382 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
dom 𝐴 = (𝐴𝐴)
 
Theoremunixpm 4880* The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)
(∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) = (𝐴𝐵))
 
Theoremunixp0im 4881 The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)
((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
 
Theoremcnvexg 4882 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
(𝐴𝑉𝐴 ∈ V)
 
Theoremcnvex 4883 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)
𝐴 ∈ V       𝐴 ∈ V
 
Theoremrelcnvexb 4884 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
(Rel 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))
 
Theoremressn 4885 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
 
Theoremcnviinm 4886* The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
(∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
 
Theoremcnvpom 4887* The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
(∃𝑥 𝑥𝐴 → (𝑅 Po 𝐴𝑅 Po 𝐴))
 
Theoremcnvsom 4888* The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)
(∃𝑥 𝑥𝐴 → (𝑅 Or 𝐴𝑅 Or 𝐴))
 
Theoremcoexg 4889 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theoremcoex 4890 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V
 
Theoremxpcom 4891* Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)
(∃𝑥 𝑥𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
 
2.6.7  Definite description binder (inverted iota)
 
Syntaxcio 4892 Extend class notation with Russell's definition description binder (inverted iota).
class (℩𝑥𝜑)
 
Theoremiotajust 4893* Soundness justification theorem for df-iota 4894. (Contributed by Andrew Salmon, 29-Jun-2011.)
{𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
 
Definitiondf-iota 4894* Define Russell's definition description binder, which can be read as "the unique 𝑥 such that 𝜑," where 𝜑 ordinarily contains 𝑥 as a free variable. Our definition is meaningful only when there is exactly one 𝑥 such that 𝜑 is true (see iotaval 4905); otherwise, it evaluates to the empty set (see iotanul 4909). Russell used the inverted iota symbol to represent the binder.

Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use iotacl 4917 (for unbounded iota). This can be easier than applying a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF.

(Contributed by Andrew Salmon, 30-Jun-2011.)

(℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
 
Theoremdfiota2 4895* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
(℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
 
Theoremnfiota1 4896 Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥(℩𝑥𝜑)
 
Theoremnfiotadxy 4897* Deduction version of nfiotaxy 4898. (Contributed by Jim Kingdon, 21-Dec-2018.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥(℩𝑦𝜓))
 
Theoremnfiotaxy 4898* Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.)
𝑥𝜑       𝑥(℩𝑦𝜑)
 
Theoremcbviota 4899 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝑦𝜑    &   𝑥𝜓       (℩𝑥𝜑) = (℩𝑦𝜓)
 
Theoremcbviotav 4900* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))       (℩𝑥𝜑) = (℩𝑦𝜓)
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