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Theorem drnfc1 2435
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
drnfc1  |-  ( A. x  x  =  y  ->  ( F/_ x A  <->  F/_ y B ) )

Proof of Theorem drnfc1
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 drnfc1.1 . . . . 5  |-  ( A. x  x  =  y  ->  A  =  B )
21eleq2d 2350 . . . 4  |-  ( A. x  x  =  y  ->  ( w  e.  A  <->  w  e.  B ) )
32drnf1 1909 . . 3  |-  ( A. x  x  =  y  ->  ( F/ x  w  e.  A  <->  F/ y  w  e.  B )
)
43dral2 1906 . 2  |-  ( A. x  x  =  y  ->  ( A. w F/ x  w  e.  A  <->  A. w F/ y  w  e.  B ) )
5 df-nfc 2408 . 2  |-  ( F/_ x A  <->  A. w F/ x  w  e.  A )
6 df-nfc 2408 . 2  |-  ( F/_ y B  <->  A. w F/ y  w  e.  B )
74, 5, 63bitr4g 279 1  |-  ( A. x  x  =  y  ->  ( F/_ x A  <->  F/_ y B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406
This theorem is referenced by:  nfabd2  2437  nfcvb  4205  nfriotad  6313
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-an 360  df-ex 1529  df-nf 1532  df-cleq 2276  df-clel 2279  df-nfc 2408
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