P. 22 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2101-2200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eleq2 2101	Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ( C e. A <-> C e. B ) )
Theorem	eleq12 2102	Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
|- ( ( A = B /\ C = D ) -> ( A e. C <-> B e. D ) )
Theorem	eleq1i 2103	Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- A = B   =>    |- ( A e. C <-> B e. C )
Theorem	eleq2i 2104	Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- A = B   =>    |- ( C e. A <-> C e. B )
Theorem	eleq12i 2105	Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
|- A = B   &    |- C = D   =>    |- ( A e. C <-> B e. D )
Theorem	eleq1d 2106	Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A e. C <-> B e. C ) )
Theorem	eleq2d 2107	Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C e. A <-> C e. B ) )
Theorem	eleq12d 2108	Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A e. C <-> B e. D ) )
Theorem	eleq1a 2109	A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
|- ( A e. B -> ( C = A -> C e. B ) )
Theorem	eqeltri 2110	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- B e. C   =>    |- A e. C
Theorem	eqeltrri 2111	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- A e. C   =>    |- B e. C
Theorem	eleqtri 2112	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
|- A e. B   &    |- B = C   =>    |- A e. C
Theorem	eleqtrri 2113	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
|- A e. B   &    |- C = B   =>    |- A e. C
Theorem	eqeltrd 2114	Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
|- ( ph -> A = B )   &    |- ( ph -> B e. C )   =>    |- ( ph -> A e. C )
Theorem	eqeltrrd 2115	Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
|- ( ph -> A = B )   &    |- ( ph -> A e. C )   =>    |- ( ph -> B e. C )
Theorem	eleqtrd 2116	Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
|- ( ph -> A e. B )   &    |- ( ph -> B = C )   =>    |- ( ph -> A e. C )
Theorem	eleqtrrd 2117	Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
|- ( ph -> A e. B )   &    |- ( ph -> C = B )   =>    |- ( ph -> A e. C )
Theorem	3eltr3i 2118	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
|- A e. B   &    |- A = C   &    |- B = D   =>    |- C e. D
Theorem	3eltr4i 2119	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
|- A e. B   &    |- C = A   &    |- D = B   =>    |- C e. D
Theorem	3eltr3d 2120	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
|- ( ph -> A e. B )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> C e. D )
Theorem	3eltr4d 2121	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
|- ( ph -> A e. B )   &    |- ( ph -> C = A )   &    |- ( ph -> D = B )   =>    |- ( ph -> C e. D )
Theorem	3eltr3g 2122	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
|- ( ph -> A e. B )   &    |- A = C   &    |- B = D   =>    |- ( ph -> C e. D )
Theorem	3eltr4g 2123	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
|- ( ph -> A e. B )   &    |- C = A   &    |- D = B   =>    |- ( ph -> C e. D )
Theorem	syl5eqel 2124	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- A = B   &    |- ( ph -> B e. C )   =>    |- ( ph -> A e. C )
Theorem	syl5eqelr 2125	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- B = A   &    |- ( ph -> B e. C )   =>    |- ( ph -> A e. C )
Theorem	syl5eleq 2126	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- A e. B   &    |- ( ph -> B = C )   =>    |- ( ph -> A e. C )
Theorem	syl5eleqr 2127	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- A e. B   &    |- ( ph -> C = B )   =>    |- ( ph -> A e. C )
Theorem	syl6eqel 2128	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- ( ph -> A = B )   &    |- B e. C   =>    |- ( ph -> A e. C )
Theorem	syl6eqelr 2129	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- ( ph -> B = A )   &    |- B e. C   =>    |- ( ph -> A e. C )
Theorem	syl6eleq 2130	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- ( ph -> A e. B )   &    |- B = C   =>    |- ( ph -> A e. C )
Theorem	syl6eleqr 2131	A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
|- ( ph -> A e. B )   &    |- C = B   =>    |- ( ph -> A e. C )
Theorem	eleq2s 2132	Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( A e. B -> ph )   &    |- C = B   =>    |- ( A e. C -> ph )
Theorem	eqneltrd 2133	If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A = B )   &    |- ( ph -> -. B e. C )   =>    |- ( ph -> -. A e. C )
Theorem	eqneltrrd 2134	If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A = B )   &    |- ( ph -> -. A e. C )   =>    |- ( ph -> -. B e. C )
Theorem	neleqtrd 2135	If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> -. C e. A )   &    |- ( ph -> A = B )   =>    |- ( ph -> -. C e. B )
Theorem	neleqtrrd 2136	If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> -. C e. B )   &    |- ( ph -> A = B )   =>    |- ( ph -> -. C e. A )
Theorem	cleqh 2137*	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2201. (Contributed by NM, 5-Aug-1993.)
|- ( y e. A -> A. x y e. A )   &    |- ( y e. B -> A. x y e. B )   =>    |- ( A = B <-> A. x ( x e. A <-> x e. B ) )
Theorem	nelneq 2138	A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
|- ( ( A e. C /\ -. B e. C ) -> -. A = B )
Theorem	nelneq2 2139	A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
|- ( ( A e. B /\ -. A e. C ) -> -. B = C )
Theorem	eqsb3lem 2140*	Lemma for eqsb3 2141. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( [ x / y ] y = A <-> x = A )
Theorem	eqsb3 2141*	Substitution applied to an atomic wff (class version of equsb3 1825). (Contributed by Rodolfo Medina, 28-Apr-2010.)
|- ( [ x / y ] y = A <-> x = A )
Theorem	clelsb3 2142*	Substitution applied to an atomic wff (class version of elsb3 1852). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( [ x / y ] y e. A <-> x e. A )
Theorem	clelsb4 2143*	Substitution applied to an atomic wff (class version of elsb4 1853). (Contributed by Jim Kingdon, 22-Nov-2018.)
|- ( [ x / y ] A e. y <-> A e. x )
Theorem	hbxfreq 2144	A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1361 for equivalence version. (Contributed by NM, 21-Aug-2007.)
|- A = B   &    |- ( y e. B -> A. x y e. B )   =>    |- ( y e. A -> A. x y e. A )
Theorem	hblem 2145*	Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
|- ( y e. A -> A. x y e. A )   =>    |- ( z e. A -> A. x z e. A )
Theorem	abeq2 2146*	Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that abbi 2151 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable. Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable ph (that has a free variable x) to a theorem with a class variable A, we substitute x e. A for ph throughout and simplify, where A is a new class variable not already in the wff. Conversely, to convert a theorem with a class variable A to one with ph, we substitute { x | ph } for A throughout and simplify, where x and ph are new set and wff variables not already in the wff. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 5-Aug-1993.)
|- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
Theorem	abeq1 2147*	Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
|- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) )
Theorem	abeq2i 2148	Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)
|- A = { x | ph }   =>    |- ( x e. A <-> ph )
Theorem	abeq1i 2149	Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)
|- { x | ph } = A   =>    |- ( ph <-> x e. A )
Theorem	abeq2d 2150	Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
|- ( ph -> A = { x | ps } )   =>    |- ( ph -> ( x e. A <-> ps ) )
Theorem	abbi 2151	Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)
|- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )
Theorem	abbi2i 2152*	Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)
|- ( x e. A <-> ph )   =>    |- A = { x | ph }
Theorem	abbii 2153	Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   =>    |- { x | ph } = { x | ps }
Theorem	abbid 2154	Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> { x | ps } = { x | ch } )
Theorem	abbidv 2155*	Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> { x | ps } = { x | ch } )
Theorem	abbi2dv 2156*	Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
|- ( ph -> ( x e. A <-> ps ) )   =>    |- ( ph -> A = { x | ps } )
Theorem	abbi1dv 2157*	Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
|- ( ph -> ( ps <-> x e. A ) )   =>    |- ( ph -> { x | ps } = A )
Theorem	abid2 2158*	A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
|- { x | x e. A } = A
Theorem	sb8ab 2159	Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
|- F/ y ph   =>    |- { x | ph } = { y | [ y / x ] ph }
Theorem	cbvab 2160	Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- { x | ph } = { y | ps }
Theorem	cbvabv 2161*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- { x | ph } = { y | ps }
Theorem	clelab 2162*	Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
|- ( A e. { x | ph } <-> E. x ( x = A /\ ph ) )
Theorem	clabel 2163*	Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
|- ( { x | ph } e. A <-> E. y ( y e. A /\ A. x ( x e. y <-> ph ) ) )
Theorem	sbab 2164*	The right-hand side of the second equality is a way of representing proper substitution of y for x into a class variable. (Contributed by NM, 14-Sep-2003.)
|- ( x = y -> A = { z | [ y / x ] z e. A } )
2.1.3  Class form not-free predicate
Syntax	wnfc 2165	Extend wff definition to include the not-free predicate for classes.
wff F/_ x A
Theorem	nfcjust 2166*	Justification theorem for df-nfc 2167. (Contributed by Mario Carneiro, 13-Oct-2016.)
|- ( A. y F/ x y e. A <-> A. z F/ x z e. A )
Definition	df-nfc 2167*	Define the not-free predicate for classes. This is read " x is not free in A". Not-free means that the value of x cannot affect the value of A, e.g., any occurrence of x in A is effectively bound by a quantifier or something that expands to one (such as "there exists at most one"). It is defined in terms of the not-free predicate df-nf 1350 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( F/_ x A <-> A. y F/ x y e. A )
Theorem	nfci 2168*	Deduce that a class A does not have x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x y e. A   =>    |- F/_ x A
Theorem	nfcii 2169*	Deduce that a class A does not have x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( y e. A -> A. x y e. A )   =>    |- F/_ x A
Theorem	nfcr 2170*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( F/_ x A -> F/ x y e. A )
Theorem	nfcrii 2171*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A   =>    |- ( y e. A -> A. x y e. A )
Theorem	nfcri 2172*	Consequence of the not-free predicate. (Note that unlike nfcr 2170, this does not require y and A to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A   =>    |- F/ x y e. A
Theorem	nfcd 2173*	Deduce that a class A does not have x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ y ph   &    |- ( ph -> F/ x y e. A )   =>    |- ( ph -> F/_ x A )
Theorem	nfceqi 2174	Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- A = B   =>    |- ( F/_ x A <-> F/_ x B )
Theorem	nfcxfr 2175	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- A = B   &    |- F/_ x B   =>    |- F/_ x A
Theorem	nfcxfrd 2176	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- A = B   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/_ x A )
Theorem	nfceqdf 2177	An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- F/ x ph   &    |- ( ph -> A = B )   =>    |- ( ph -> ( F/_ x A <-> F/_ x B ) )
Theorem	nfcv 2178*	If x is disjoint from A, then x is not free in A. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A
Theorem	nfcvd 2179*	If x is disjoint from A, then x is not free in A. (Contributed by Mario Carneiro, 7-Oct-2016.)
|- ( ph -> F/_ x A )
Theorem	nfab1 2180	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x { x | ph }
Theorem	nfnfc1 2181	x is bound in F/_ x A. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x F/_ x A
Theorem	nfab 2182	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   =>    |- F/_ x { y | ph }
Theorem	nfaba1 2183	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- F/_ x { y | A. x ph }
Theorem	nfnfc 2184	Hypothesis builder for F/_ y A. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A   =>    |- F/ x F/_ y A
Theorem	nfeq 2185	Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/ x A = B
Theorem	nfel 2186	Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/ x A e. B
Theorem	nfeq1 2187*	Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   =>    |- F/ x A = B
Theorem	nfel1 2188*	Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   =>    |- F/ x A e. B
Theorem	nfeq2 2189*	Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
|- F/_ x B   =>    |- F/ x A = B
Theorem	nfel2 2190*	Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
|- F/_ x B   =>    |- F/ x A e. B
Theorem	nfcrd 2191*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph -> F/_ x A )   =>    |- ( ph -> F/ x y e. A )
Theorem	nfeqd 2192	Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
|- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/ x A = B )
Theorem	nfeld 2193	Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
|- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/ x A e. B )
Theorem	drnfc1 2194	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
|- ( A. x x = y -> A = B )   =>    |- ( A. x x = y -> ( F/_ x A <-> F/_ y B ) )
Theorem	drnfc2 2195	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
|- ( A. x x = y -> A = B )   =>    |- ( A. x x = y -> ( F/_ z A <-> F/_ z B ) )
Theorem	nfabd 2196	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/_ x { y | ps } )
Theorem	dvelimdc 2197	Deduction form of dvelimc 2198. (Contributed by Mario Carneiro, 8-Oct-2016.)
|- F/ x ph   &    |- F/ z ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/_ z B )   &    |- ( ph -> ( z = y -> A = B ) )   =>    |- ( ph -> ( -. A. x x = y -> F/_ x B ) )
Theorem	dvelimc 2198	Version of dvelim 1893 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
|- F/_ x A   &    |- F/_ z B   &    |- ( z = y -> A = B )   =>    |- ( -. A. x x = y -> F/_ x B )
Theorem	nfcvf 2199	If x and y are distinct, then x is not free in y. (Contributed by Mario Carneiro, 8-Oct-2016.)
|- ( -. A. x x = y -> F/_ x y )
Theorem	nfcvf2 2200	If x and y are distinct, then y is not free in x. (Contributed by Mario Carneiro, 5-Dec-2016.)
|- ( -. A. x x = y -> F/_ y x )