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Theorem drnfc1 2299
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 2210 . . . 4  |-  ( A. x  x  =  y  ->  ( w  e.  A  <->  w  e.  B ) )
32drnf1 1712 . . 3  |-  ( A. x  x  =  y  ->  ( F/ x  w  e.  A  <->  F/ y  w  e.  B )
)
43dral2 1710 . 2  |-  ( A. x  x  =  y  ->  ( A. w F/ x  w  e.  A  <->  A. w F/ y  w  e.  B ) )
5 df-nfc 2271 . 2  |-  ( F/_ x A  <->  A. w F/ x  w  e.  A )
6 df-nfc 2271 . 2  |-  ( F/_ y B  <->  A. w F/ y  w  e.  B )
74, 5, 63bitr4g 222 1  |-  ( A. x  x  =  y  ->  ( F/_ x A  <->  F/_ y B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1330    = wceq 1332   F/wnf 1437    e. wcel 1481   F/_wnfc 2269
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-clel 2136  df-nfc 2271
This theorem is referenced by: (None)
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