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Theorem drnfc1 2239
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 2152 . . . 4  |-  ( A. x  x  =  y  ->  ( w  e.  A  <->  w  e.  B ) )
32drnf1 1663 . . 3  |-  ( A. x  x  =  y  ->  ( F/ x  w  e.  A  <->  F/ y  w  e.  B )
)
43dral2 1661 . 2  |-  ( A. x  x  =  y  ->  ( A. w F/ x  w  e.  A  <->  A. w F/ y  w  e.  B ) )
5 df-nfc 2212 . 2  |-  ( F/_ x A  <->  A. w F/ x  w  e.  A )
6 df-nfc 2212 . 2  |-  ( F/_ y B  <->  A. w F/ y  w  e.  B )
74, 5, 63bitr4g 221 1  |-  ( A. x  x  =  y  ->  ( F/_ x A  <->  F/_ y B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283    = wceq 1285   F/wnf 1390    e. wcel 1434   F/_wnfc 2210
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-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-cleq 2076  df-clel 2079  df-nfc 2212
This theorem is referenced by: (None)
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