Theorem nfccdeq 2953

Description: Variation of nfcdeq 2952 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
nfccdeq.1	|- F/_ x A
nfccdeq.2	|- CondEq ( x = y -> A = B )

Assertion

Ref	Expression
nfccdeq	|- A = B

Distinct variable groups:   x, B   y, A

Allowed substitution hints:   A( x)   B( y)

Proof of Theorem nfccdeq

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfccdeq.1	. . . 4 |- F/_ x A
2	1	nfcri 2306	. . 3 |- F/ x z e. A
3		equid 1694	. . . . 5 |- z = z
4	3	cdeqth 2942	. . . 4 |- CondEq ( x = y -> z = z )
5		nfccdeq.2	. . . 4 |- CondEq ( x = y -> A = B )
6	4, 5	cdeqel 2951	. . 3 |- CondEq ( x = y -> ( z e. A <-> z e. B ) )
7	2, 6	nfcdeq 2952	. 2 |- ( z e. A <-> z e. B )
8	7	eqriv 2167	1 |- A = B

Syntax hints:   = wceq 1348   e. wcel 2141   F/_wnfc 2299  CondEqwcdeq 2938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-cdeq 2939

This theorem is referenced by: (None)