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	syl5eleq 2201	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- A e. B   &    |- ( ph -> B = C )   =>    |- ( ph -> A e. C )
Theorem	syl5eleqr 2202	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- A e. B   &    |- ( ph -> C = B )   =>    |- ( ph -> A e. C )
Theorem	syl6eqel 2203	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- ( ph -> A = B )   &    |- B e. C   =>    |- ( ph -> A e. C )
Theorem	syl6eqelr 2204	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- ( ph -> B = A )   &    |- B e. C   =>    |- ( ph -> A e. C )
Theorem	syl6eleq 2205	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
|- ( ph -> A e. B )   &    |- B = C   =>    |- ( ph -> A e. C )
Theorem	syl6eleqr 2206	A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
|- ( ph -> A e. B )   &    |- C = B   =>    |- ( ph -> A e. C )
Theorem	eleq2s 2207	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 2208	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 2209	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 2210	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 2211	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 2212*	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2277. (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 2213	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 2214	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 2215*	Lemma for eqsb3 2216. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( [ y / x ] x = A <-> y = A )
Theorem	eqsb3 2216*	Substitution applied to an atomic wff (class version of equsb3 1898). (Contributed by Rodolfo Medina, 28-Apr-2010.)
|- ( [ y / x ] x = A <-> y = A )
Theorem	clelsb3 2217*	Substitution applied to an atomic wff (class version of elsb3 1925). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( [ y / x ] x e. A <-> y e. A )
Theorem	clelsb4 2218*	Substitution applied to an atomic wff (class version of elsb4 1926). (Contributed by Jim Kingdon, 22-Nov-2018.)
|- ( [ y / x ] A e. x <-> A e. y )
Theorem	hbxfreq 2219	A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1429 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 2220*	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 2221*	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 2226 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 2222*	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 2223	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 2224	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 2225	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 2226	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 2227*	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 2228	Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   =>    |- { x | ph } = { x | ps }
Theorem	abbid 2229	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 2230*	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 2231*	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 2232*	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 2233*	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 2234	Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
|- F/ y ph   =>    |- { x | ph } = { y | [ y / x ] ph }
Theorem	cbvab 2235	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 2236*	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 2237*	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 2238*	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 2239*	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 2240	Extend wff definition to include the not-free predicate for classes.
wff F/_ x A
Theorem	nfcjust 2241*	Justification theorem for df-nfc 2242. (Contributed by Mario Carneiro, 13-Oct-2016.)
|- ( A. y F/ x y e. A <-> A. z F/ x z e. A )
Definition	df-nfc 2242*	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 1418 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 2243*	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 2244*	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 2245*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( F/_ x A -> F/ x y e. A )
Theorem	nfcrii 2246*	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 2247*	Consequence of the not-free predicate. (Note that unlike nfcr 2245, 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 2248*	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 2249	Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- A = B   =>    |- ( F/_ x A <-> F/_ x B )
Theorem	nfcxfr 2250	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 2251	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 2252	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 2253*	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 2254*	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 2255	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x { x | ph }
Theorem	nfnfc1 2256	x is bound in F/_ x A. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x F/_ x A
Theorem	clelsb3f 2257	Substitution applied to an atomic wff (class version of elsb3 1925). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
|- F/_ x A   =>    |- ( [ y / x ] x e. A <-> y e. A )
Theorem	nfab 2258	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   =>    |- F/_ x { y | ph }
Theorem	nfaba1 2259	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- F/_ x { y | A. x ph }
Theorem	nfnfc 2260	Hypothesis builder for F/_ y A. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A   =>    |- F/ x F/_ y A
Theorem	nfeq 2261	Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/ x A = B
Theorem	nfel 2262	Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/ x A e. B
Theorem	nfeq1 2263*	Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   =>    |- F/ x A = B
Theorem	nfel1 2264*	Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   =>    |- F/ x A e. B
Theorem	nfeq2 2265*	Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
|- F/_ x B   =>    |- F/ x A = B
Theorem	nfel2 2266*	Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
|- F/_ x B   =>    |- F/ x A e. B
Theorem	nfcrd 2267*	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 2268	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 2269	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 2270	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 2271	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 2272	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 2273	Deduction form of dvelimc 2274. (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 2274	Version of dvelim 1966 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 2275	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 2276	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 )
Theorem	cleqf 2277	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2212. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A = B <-> A. x ( x e. A <-> x e. B ) )
Theorem	abid2f 2278	A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/_ x A   =>    |- { x | x e. A } = A
Theorem	sbabel 2279*	Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/_ x A   =>    |- ( [ y / x ] { z | ph } e. A <-> { z | [ y / x ] ph } e. A )
2.1.4  Negated equality and membership
2.1.4.1  Negated equality
Syntax	wne 2280	Extend wff notation to include inequality.
wff A =/= B
Definition	df-ne 2281	Define inequality. (Contributed by NM, 5-Aug-1993.)
|- ( A =/= B <-> -. A = B )
Theorem	neii 2282	Inference associated with df-ne 2281. (Contributed by BJ, 7-Jul-2018.)
|- A =/= B   =>    |- -. A = B
Theorem	neir 2283	Inference associated with df-ne 2281. (Contributed by BJ, 7-Jul-2018.)
|- -. A = B   =>    |- A =/= B
Theorem	nner 2284	Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
|- ( A = B -> -. A =/= B )
Theorem	nnedc 2285	Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
|- (DECID A = B -> ( -. A =/= B <-> A = B ) )
Theorem	dcned 2286	Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
|- ( ph -> DECID A = B )   =>    |- ( ph -> DECID A =/= B )
Theorem	neqned 2287	If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2301. One-way deduction form of df-ne 2281. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2330. (Revised by Wolf Lammen, 22-Nov-2019.)
|- ( ph -> -. A = B )   =>    |- ( ph -> A =/= B )
Theorem	neqne 2288	From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( -. A = B -> A =/= B )
Theorem	neirr 2289	No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
|- -. A =/= A
Theorem	eqneqall 2290	A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
|- ( A = B -> ( A =/= B -> ph ) )
Theorem	dcne 2291	Decidable equality expressed in terms of =/=. Basically the same as df-dc 803. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- (DECID A = B <-> ( A = B \/ A =/= B ) )
Theorem	nonconne 2292	Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
|- -. ( A = B /\ A =/= B )
Theorem	neeq1 2293	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
|- ( A = B -> ( A =/= C <-> B =/= C ) )
Theorem	neeq2 2294	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
|- ( A = B -> ( C =/= A <-> C =/= B ) )
Theorem	neeq1i 2295	Inference for inequality. (Contributed by NM, 29-Apr-2005.)
|- A = B   =>    |- ( A =/= C <-> B =/= C )
Theorem	neeq2i 2296	Inference for inequality. (Contributed by NM, 29-Apr-2005.)
|- A = B   =>    |- ( C =/= A <-> C =/= B )
Theorem	neeq12i 2297	Inference for inequality. (Contributed by NM, 24-Jul-2012.)
|- A = B   &    |- C = D   =>    |- ( A =/= C <-> B =/= D )
Theorem	neeq1d 2298	Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A =/= C <-> B =/= C ) )
Theorem	neeq2d 2299	Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C =/= A <-> C =/= B ) )
Theorem	neeq12d 2300	Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A =/= C <-> B =/= D ) )