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Theorem drnfc2 2276
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypothesis
Ref Expression
drnfc1.1  |-  ( A. x  x  =  y  ->  A  =  B )
Assertion
Ref Expression
drnfc2  |-  ( A. x  x  =  y  ->  ( F/_ z A  <->  F/_ z B ) )

Proof of Theorem drnfc2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 drnfc1.1 . . . . 5  |-  ( A. x  x  =  y  ->  A  =  B )
21eleq2d 2187 . . . 4  |-  ( A. x  x  =  y  ->  ( w  e.  A  <->  w  e.  B ) )
32drnf2 1697 . . 3  |-  ( A. x  x  =  y  ->  ( F/ z  w  e.  A  <->  F/ z  w  e.  B )
)
43dral2 1694 . 2  |-  ( A. x  x  =  y  ->  ( A. w F/ z  w  e.  A  <->  A. w F/ z  w  e.  B ) )
5 df-nfc 2247 . 2  |-  ( F/_ z A  <->  A. w F/ z  w  e.  A )
6 df-nfc 2247 . 2  |-  ( F/_ z B  <->  A. w F/ z  w  e.  B )
74, 5, 63bitr4g 222 1  |-  ( A. x  x  =  y  ->  ( F/_ z A  <->  F/_ z B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1314    = wceq 1316   F/wnf 1421    e. wcel 1465   F/_wnfc 2245
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-cleq 2110  df-clel 2113  df-nfc 2247
This theorem is referenced by: (None)
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