Theorem nfccdeq 3433

Description: Variation of nfcdeq 3432 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 2758	. . 3 |- F/ x z e. A
3		equid 1939	. . . . 5 |- z = z
4	3	cdeqth 3422	. . . 4 |- CondEq ( x = y -> z = z )
5		nfccdeq.2	. . . 4 |- CondEq ( x = y -> A = B )
6	4, 5	cdeqel 3431	. . 3 |- CondEq ( x = y -> ( z e. A <-> z e. B ) )
7	2, 6	nfcdeq 3432	. 2 |- ( z e. A <-> z e. B )
8	7	eqriv 2619	1 |- A = B

Syntax hints:   = wceq 1483   e. wcel 1990   F/_wnfc 2751  CondEqwcdeq 3418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618  df-nfc 2753  df-cdeq 3419

This theorem is referenced by: (None)