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Theorem nfccdeq 2814
Description: Variation of nfcdeq 2813 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 2214 . . 3  |-  F/ x  z  e.  A
3 equid 1630 . . . . 5  |-  z  =  z
43cdeqth 2803 . . . 4  |- CondEq ( x  =  y  ->  z  =  z )
5 nfccdeq.2 . . . 4  |- CondEq ( x  =  y  ->  A  =  B )
64, 5cdeqel 2812 . . 3  |- CondEq ( x  =  y  ->  (
z  e.  A  <->  z  e.  B ) )
72, 6nfcdeq 2813 . 2  |-  ( z  e.  A  <->  z  e.  B )
87eqriv 2079 1  |-  A  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   F/_wnfc 2207  CondEqwcdeq 2799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078  df-nfc 2209  df-cdeq 2800
This theorem is referenced by: (None)
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