P. 24 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2301-2400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eqneltrrd 2301	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 2302	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 2303	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 2304*	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2372. (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 2305	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 2306	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 2307*	Lemma for eqsb1 2308. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( [ y / x ] x = A <-> y = A )
Theorem	eqsb1 2308*	Substitution for the left-hand side in an equality. Class version of equsb3 1978. (Contributed by Rodolfo Medina, 28-Apr-2010.)
|- ( [ y / x ] x = A <-> y = A )
Theorem	clelsb1 2309*	Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2182). (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 2310*	Substitution for the second argument of the membership predicate in an atomic formula (class version of elsb2 2183). (Contributed by Jim Kingdon, 22-Nov-2018.)
|- ( [ y / x ] A e. x <-> A e. y )
Theorem	hbxfreq 2311	A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1494 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 2312*	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 2313*	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 2318 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 2314*	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 2315	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 2316	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 2317	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 2318	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 2319*	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 2320	Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   =>    |- { x | ph } = { x | ps }
Theorem	abbid 2321	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 2322*	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 2323*	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 2324*	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 2325*	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 2326	Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
|- F/ y ph   =>    |- { x | ph } = { y | [ y / x ] ph }
Theorem	cbvabw 2327*	Version of cbvab 2328 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- { x | ph } = { y | ps }
Theorem	cbvab 2328	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 2329*	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 2330*	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 2331*	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 2332*	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 } )
2.1.2.1  Elementary properties of class abstractions
Theorem	eqabdv 2333*	Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Revised by Wolf Lammen, 6-May-2023.)
|- ( ph -> ( x e. A <-> ps ) )   =>    |- ( ph -> A = { x | ps } )
2.1.3  Class form not-free predicate
Syntax	wnfc 2334	Extend wff definition to include the not-free predicate for classes.
wff F/_ x A
Theorem	nfcjust 2335*	Justification theorem for df-nfc 2336. (Contributed by Mario Carneiro, 13-Oct-2016.)
|- ( A. y F/ x y e. A <-> A. z F/ x z e. A )
Definition	df-nfc 2336*	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 1483 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 2337*	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 2338*	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 2339*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( F/_ x A -> F/ x y e. A )
Theorem	nfcrii 2340*	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 2341*	Consequence of the not-free predicate. (Note that unlike nfcr 2339, 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 2342*	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 2343	Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- A = B   =>    |- ( F/_ x A <-> F/_ x B )
Theorem	nfcxfr 2344	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 2345	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 2346	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 2347*	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 2348*	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 2349	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x { x | ph }
Theorem	nfnfc1 2350	x is bound in F/_ x A. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x F/_ x A
Theorem	clelsb1f 2351	Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2182). (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 2352	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   =>    |- F/_ x { y | ph }
Theorem	nfaba1 2353	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- F/_ x { y | A. x ph }
Theorem	nfnfc 2354	Hypothesis builder for F/_ y A. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A   =>    |- F/ x F/_ y A
Theorem	nfeq 2355	Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/ x A = B
Theorem	nfel 2356	Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/ x A e. B
Theorem	nfeq1 2357*	Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   =>    |- F/ x A = B
Theorem	nfel1 2358*	Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
|- F/_ x A   =>    |- F/ x A e. B
Theorem	nfeq2 2359*	Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
|- F/_ x B   =>    |- F/ x A = B
Theorem	nfel2 2360*	Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
|- F/_ x B   =>    |- F/ x A e. B
Theorem	nfcrd 2361*	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 2362	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 2363	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 2364	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 2365	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	nfabdw 2366*	Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2367 with a disjoint variable condition. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by GG, 10-Jan-2024.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/_ x { y | ps } )
Theorem	nfabd 2367	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 2368	Deduction form of dvelimc 2369. (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 2369	Version of dvelim 2044 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 2370	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 2371	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 2372	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2304. (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 2373	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 2374*	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 2375	Extend wff notation to include inequality.
wff A =/= B
Definition	df-ne 2376	Define inequality. (Contributed by NM, 5-Aug-1993.)
|- ( A =/= B <-> -. A = B )
Theorem	neii 2377	Inference associated with df-ne 2376. (Contributed by BJ, 7-Jul-2018.)
|- A =/= B   =>    |- -. A = B
Theorem	neir 2378	Inference associated with df-ne 2376. (Contributed by BJ, 7-Jul-2018.)
|- -. A = B   =>    |- A =/= B
Theorem	nner 2379	Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
|- ( A = B -> -. A =/= B )
Theorem	nnedc 2380	Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
|- (DECID A = B -> ( -. A =/= B <-> A = B ) )
Theorem	dcned 2381	Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
|- ( ph -> DECID A = B )   =>    |- ( ph -> DECID A =/= B )
Theorem	neqned 2382	If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2396. One-way deduction form of df-ne 2376. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2425. (Revised by Wolf Lammen, 22-Nov-2019.)
|- ( ph -> -. A = B )   =>    |- ( ph -> A =/= B )
Theorem	neqne 2383	From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( -. A = B -> A =/= B )
Theorem	neirr 2384	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 2385	A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
|- ( A = B -> ( A =/= B -> ph ) )
Theorem	dcne 2386	Decidable equality expressed in terms of =/=. Basically the same as df-dc 836. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- (DECID A = B <-> ( A = B \/ A =/= B ) )
Theorem	nonconne 2387	Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
|- -. ( A = B /\ A =/= B )
Theorem	neeq1 2388	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
|- ( A = B -> ( A =/= C <-> B =/= C ) )
Theorem	neeq2 2389	Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
|- ( A = B -> ( C =/= A <-> C =/= B ) )
Theorem	neeq1i 2390	Inference for inequality. (Contributed by NM, 29-Apr-2005.)
|- A = B   =>    |- ( A =/= C <-> B =/= C )
Theorem	neeq2i 2391	Inference for inequality. (Contributed by NM, 29-Apr-2005.)
|- A = B   =>    |- ( C =/= A <-> C =/= B )
Theorem	neeq12i 2392	Inference for inequality. (Contributed by NM, 24-Jul-2012.)
|- A = B   &    |- C = D   =>    |- ( A =/= C <-> B =/= D )
Theorem	neeq1d 2393	Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A =/= C <-> B =/= C ) )
Theorem	neeq2d 2394	Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C =/= A <-> C =/= B ) )
Theorem	neeq12d 2395	Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A =/= C <-> B =/= D ) )
Theorem	neneqd 2396	Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph -> A =/= B )   =>    |- ( ph -> -. A = B )
Theorem	neneq 2397	From inequality to non-equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A =/= B -> -. A = B )
Theorem	eqnetri 2398	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- A = B   &    |- B =/= C   =>    |- A =/= C
Theorem	eqnetrd 2399	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- ( ph -> A = B )   &    |- ( ph -> B =/= C )   =>    |- ( ph -> A =/= C )
Theorem	eqnetrri 2400	Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
|- A = B   &    |- A =/= C   =>    |- B =/= C