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Theorem nfccdeq 3002
Description: Variation of nfcdeq 3001 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.
StepHypRef Expression
1 nfccdeq.1 . . . 4  |-  F/_ x A
21nfcri 2426 . . 3  |-  F/ x  z  e.  A
3 equid 1662 . . . . 5  |-  z  =  z
43cdeqth 2991 . . . 4  |- CondEq ( x  =  y  ->  z  =  z )
5 nfccdeq.2 . . . 4  |- CondEq ( x  =  y  ->  A  =  B )
64, 5cdeqel 3000 . . 3  |- CondEq ( x  =  y  ->  (
z  e.  A  <->  z  e.  B ) )
72, 6nfcdeq 3001 . 2  |-  ( z  e.  A  <->  z  e.  B )
87eqriv 2293 1  |-  A  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   F/_wnfc 2419  CondEqwcdeq 2987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-cleq 2289  df-clel 2292  df-nfc 2421  df-cdeq 2988
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