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Theorem nfccdeq 2989
Description: Variation of nfcdeq 2988 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 2413 . . 3  |-  F/ x  z  e.  A
3 equid 1644 . . . . 5  |-  z  =  z
43cdeqth 2978 . . . 4  |- CondEq ( x  =  y  ->  z  =  z )
5 nfccdeq.2 . . . 4  |- CondEq ( x  =  y  ->  A  =  B )
64, 5cdeqel 2987 . . 3  |- CondEq ( x  =  y  ->  (
z  e.  A  <->  z  e.  B ) )
72, 6nfcdeq 2988 . 2  |-  ( z  e.  A  <->  z  e.  B )
87eqriv 2280 1  |-  A  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   F/_wnfc 2406  CondEqwcdeq 2974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408  df-cdeq 2975
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