ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfccdeq Unicode version

Theorem nfccdeq 2953
Description: Variation of nfcdeq 2952 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 2306 . . 3  |-  F/ x  z  e.  A
3 equid 1694 . . . . 5  |-  z  =  z
43cdeqth 2942 . . . 4  |- CondEq ( x  =  y  ->  z  =  z )
5 nfccdeq.2 . . . 4  |- CondEq ( x  =  y  ->  A  =  B )
64, 5cdeqel 2951 . . 3  |- CondEq ( x  =  y  ->  (
z  e.  A  <->  z  e.  B ) )
72, 6nfcdeq 2952 . 2  |-  ( z  e.  A  <->  z  e.  B )
87eqriv 2167 1  |-  A  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1348    e. wcel 2141   F/_wnfc 2299  CondEqwcdeq 2938
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-cdeq 2939
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator