Theorem nfccdeq 2935

Description: Variation of nfcdeq 2934 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 2293	. . 3 |- F/ x z e. A
3		equid 1681	. . . . 5 |- z = z
4	3	cdeqth 2924	. . . 4 |- CondEq ( x = y -> z = z )
5		nfccdeq.2	. . . 4 |- CondEq ( x = y -> A = B )
6	4, 5	cdeqel 2933	. . 3 |- CondEq ( x = y -> ( z e. A <-> z e. B ) )
7	2, 6	nfcdeq 2934	. 2 |- ( z e. A <-> z e. B )
8	7	eqriv 2154	1 |- A = B

Syntax hints:   = wceq 1335   e. wcel 2128   F/_wnfc 2286  CondEqwcdeq 2920

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-nfc 2288  df-cdeq 2921

This theorem is referenced by: (None)