Theorem nfccdeq 2972

Description: Variation of nfcdeq 2971 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 2323	. . 3 |- F/ x z e. A
3		equid 1711	. . . . 5 |- z = z
4	3	cdeqth 2961	. . . 4 |- CondEq ( x = y -> z = z )
5		nfccdeq.2	. . . 4 |- CondEq ( x = y -> A = B )
6	4, 5	cdeqel 2970	. . 3 |- CondEq ( x = y -> ( z e. A <-> z e. B ) )
7	2, 6	nfcdeq 2971	. 2 |- ( z e. A <-> z e. B )
8	7	eqriv 2184	1 |- A = B

Syntax hints:   = wceq 1363   e. wcel 2158   F/_wnfc 2316  CondEqwcdeq 2957

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-cleq 2180  df-clel 2183  df-nfc 2318  df-cdeq 2958

This theorem is referenced by: (None)