Theorem nfccdeq 2996

Description: Variation of nfcdeq 2995 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 2342	. . 3 |- F/ x z e. A
3		equid 1724	. . . . 5 |- z = z
4	3	cdeqth 2985	. . . 4 |- CondEq ( x = y -> z = z )
5		nfccdeq.2	. . . 4 |- CondEq ( x = y -> A = B )
6	4, 5	cdeqel 2994	. . 3 |- CondEq ( x = y -> ( z e. A <-> z e. B ) )
7	2, 6	nfcdeq 2995	. 2 |- ( z e. A <-> z e. B )
8	7	eqriv 2202	1 |- A = B

Syntax hints:   = wceq 1373   e. wcel 2176   F/_wnfc 2335  CondEqwcdeq 2981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337  df-cdeq 2982

This theorem is referenced by: (None)