P. 23 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2201-2300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	3eqtr4i 2201	An inference from three chained equalities. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &    |- C = A   &    |- D = B   =>    |- C = D
Theorem	3eqtr4ri 2202	An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &    |- C = A   &    |- D = B   =>    |- D = C
Theorem	eqtrd 2203	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A = B )   &    |- ( ph -> B = C )   =>    |- ( ph -> A = C )
Theorem	eqtr2d 2204	An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
|- ( ph -> A = B )   &    |- ( ph -> B = C )   =>    |- ( ph -> C = A )
Theorem	eqtr3d 2205	An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
|- ( ph -> A = B )   &    |- ( ph -> A = C )   =>    |- ( ph -> B = C )
Theorem	eqtr4d 2206	An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
|- ( ph -> A = B )   &    |- ( ph -> C = B )   =>    |- ( ph -> A = C )
Theorem	3eqtrd 2207	A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)
|- ( ph -> A = B )   &    |- ( ph -> B = C )   &    |- ( ph -> C = D )   =>    |- ( ph -> A = D )
Theorem	3eqtrrd 2208	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A = B )   &    |- ( ph -> B = C )   &    |- ( ph -> C = D )   =>    |- ( ph -> D = A )
Theorem	3eqtr2d 2209	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
|- ( ph -> A = B )   &    |- ( ph -> C = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> A = D )
Theorem	3eqtr2rd 2210	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
|- ( ph -> A = B )   &    |- ( ph -> C = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> D = A )
Theorem	3eqtr3d 2211	A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A = B )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> C = D )
Theorem	3eqtr3rd 2212	A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)
|- ( ph -> A = B )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> D = C )
Theorem	3eqtr4d 2213	A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A = B )   &    |- ( ph -> C = A )   &    |- ( ph -> D = B )   =>    |- ( ph -> C = D )
Theorem	3eqtr4rd 2214	A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.)
|- ( ph -> A = B )   &    |- ( ph -> C = A )   &    |- ( ph -> D = B )   =>    |- ( ph -> D = C )
Theorem	eqtrid 2215	An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.)
|- A = B   &    |- ( ph -> B = C )   =>    |- ( ph -> A = C )
Theorem	eqtr2id 2216	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
|- A = B   &    |- ( ph -> B = C )   =>    |- ( ph -> C = A )
Theorem	eqtr3id 2217	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
|- B = A   &    |- ( ph -> B = C )   =>    |- ( ph -> A = C )
Theorem	eqtr3di 2218	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
|- ( ph -> A = B )   &    |- A = C   =>    |- ( ph -> B = C )
Theorem	eqtrdi 2219	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A = B )   &    |- B = C   =>    |- ( ph -> A = C )
Theorem	eqtr2di 2220	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
|- ( ph -> A = B )   &    |- B = C   =>    |- ( ph -> C = A )
Theorem	eqtr4di 2221	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A = B )   &    |- C = B   =>    |- ( ph -> A = C )
Theorem	eqtr4id 2222	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
|- A = B   &    |- ( ph -> C = B )   =>    |- ( ph -> A = C )
Theorem	sylan9eq 2223	An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A = B )   &    |- ( ps -> B = C )   =>    |- ( ( ph /\ ps ) -> A = C )
Theorem	sylan9req 2224	An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
|- ( ph -> B = A )   &    |- ( ps -> B = C )   =>    |- ( ( ph /\ ps ) -> A = C )
Theorem	sylan9eqr 2225	An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
|- ( ph -> A = B )   &    |- ( ps -> B = C )   =>    |- ( ( ps /\ ph ) -> A = C )
Theorem	3eqtr3g 2226	A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)
|- ( ph -> A = B )   &    |- A = C   &    |- B = D   =>    |- ( ph -> C = D )
Theorem	3eqtr3a 2227	A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)
|- A = B   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> C = D )
Theorem	3eqtr4g 2228	A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A = B )   &    |- C = A   &    |- D = B   =>    |- ( ph -> C = D )
Theorem	3eqtr4a 2229	A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &    |- ( ph -> C = A )   &    |- ( ph -> D = B )   =>    |- ( ph -> C = D )
Theorem	eq2tri 2230	A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
|- ( A = C -> D = F )   &    |- ( B = D -> C = G )   =>    |- ( ( A = C /\ B = F ) <-> ( B = D /\ A = G ) )
Theorem	eleq1w 2231	Weaker version of eleq1 2233 (but more general than elequ1 2145) not depending on ax-ext 2152 nor df-cleq 2163. (Contributed by BJ, 24-Jun-2019.)
|- ( x = y -> ( x e. A <-> y e. A ) )
Theorem	eleq2w 2232	Weaker version of eleq2 2234 (but more general than elequ2 2146) not depending on ax-ext 2152 nor df-cleq 2163. (Contributed by BJ, 29-Sep-2019.)
|- ( x = y -> ( A e. x <-> A e. y ) )
Theorem	eleq1 2233	Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ( A e. C <-> B e. C ) )
Theorem	eleq2 2234	Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ( C e. A <-> C e. B ) )
Theorem	eleq12 2235	Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
|- ( ( A = B /\ C = D ) -> ( A e. C <-> B e. D ) )
Theorem	eleq1i 2236	Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- A = B   =>    |- ( A e. C <-> B e. C )
Theorem	eleq2i 2237	Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- A = B   =>    |- ( C e. A <-> C e. B )
Theorem	eleq12i 2238	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 2239	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 2240	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 2241	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 2242	A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
|- ( A e. B -> ( C = A -> C e. B ) )
Theorem	eqeltri 2243	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- B e. C   =>    |- A e. C
Theorem	eqeltrri 2244	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- A e. C   =>    |- B e. C
Theorem	eleqtri 2245	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
|- A e. B   &    |- B = C   =>    |- A e. C
Theorem	eleqtrri 2246	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
|- A e. B   &    |- C = B   =>    |- A e. C
Theorem	eqeltrd 2247	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 2248	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 2249	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 2250	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 2251	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 2252	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 2253	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 2254	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 2255	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 2256	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	eqeltrid 2257	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- A = B   &    |- ( ph -> B e. C )   =>    |- ( ph -> A e. C )
Theorem	eqeltrrid 2258	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- B = A   &    |- ( ph -> B e. C )   =>    |- ( ph -> A e. C )
Theorem	eleqtrid 2259	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- A e. B   &    |- ( ph -> B = C )   =>    |- ( ph -> A e. C )
Theorem	eleqtrrid 2260	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- A e. B   &    |- ( ph -> C = B )   =>    |- ( ph -> A e. C )
Theorem	eqeltrdi 2261	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- ( ph -> A = B )   &    |- B e. C   =>    |- ( ph -> A e. C )
Theorem	eqeltrrdi 2262	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- ( ph -> B = A )   &    |- B e. C   =>    |- ( ph -> A e. C )
Theorem	eleqtrdi 2263	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- ( ph -> A e. B )   &    |- B = C   =>    |- ( ph -> A e. C )
Theorem	eleqtrrdi 2264	A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
|- ( ph -> A e. B )   &    |- C = B   =>    |- ( ph -> A e. C )
Theorem	eleq2s 2265	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 2266	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 2267	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 2268	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 2269	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 2270*	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2337. (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 2271	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 2272	A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
|- ( ( A e. B /\ -. A e. C ) -> -. B = C )
Theorem	eqsb1lem 2273*	Lemma for eqsb1 2274. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( [ y / x ] x = A <-> y = A )
Theorem	eqsb1 2274*	Substitution for the left-hand side in an equality. Class version of equsb3 1944. (Contributed by Rodolfo Medina, 28-Apr-2010.)
|- ( [ y / x ] x = A <-> y = A )
Theorem	clelsb1 2275*	Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2148). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( [ y / x ] x e. A <-> y e. A )
Theorem	clelsb2 2276*	Substitution for the second argument of the membership predicate in an atomic formula (class version of elsb2 2149). (Contributed by Jim Kingdon, 22-Nov-2018.)
|- ( [ y / x ] A e. x <-> A e. y )
Theorem	hbxfreq 2277	A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1465 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 2278*	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 2279*	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 2284 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 2280*	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 2281	Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.)
|- A = { x | ph }   =>    |- ( x e. A <-> ph )
Theorem	abeq1i 2282	Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.)
|- { x | ph } = A   =>    |- ( ph <-> x e. A )
Theorem	abeq2d 2283	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 2284	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 2285*	Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 5-Aug-1993.)
|- ( x e. A <-> ph )   =>    |- A = { x | ph }
Theorem	abbii 2286	Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   =>    |- { x | ph } = { x | ps }
Theorem	abbid 2287	Equivalent wff's yield equal class abstractions (deduction form). (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 2288*	Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> { x | ps } = { x | ch } )
Theorem	abbi2dv 2289*	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 2290*	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 2291*	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 2292	Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
|- F/ y ph   =>    |- { x | ph } = { y | [ y / x ] ph }
Theorem	cbvabw 2293*	Version of cbvab 2294 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- { x | ph } = { y | ps }
Theorem	cbvab 2294	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 2295*	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 2296*	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 2297*	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 2298*	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 2299	Extend wff definition to include the not-free predicate for classes.
wff F/_ x A
Theorem	nfcjust 2300*	Justification theorem for df-nfc 2301. (Contributed by Mario Carneiro, 13-Oct-2016.)
|- ( A. y F/ x y e. A <-> A. z F/ x z e. A )