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Theorem nfceqdf 2177
Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfceqdf.1  |-  F/ x ph
nfceqdf.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
nfceqdf  |-  ( ph  ->  ( F/_ x A  <->  F/_ x B ) )

Proof of Theorem nfceqdf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfceqdf.1 . . . 4  |-  F/ x ph
2 nfceqdf.2 . . . . 5  |-  ( ph  ->  A  =  B )
32eleq2d 2107 . . . 4  |-  ( ph  ->  ( y  e.  A  <->  y  e.  B ) )
41, 3nfbidf 1432 . . 3  |-  ( ph  ->  ( F/ x  y  e.  A  <->  F/ x  y  e.  B )
)
54albidv 1705 . 2  |-  ( ph  ->  ( A. y F/ x  y  e.  A  <->  A. y F/ x  y  e.  B ) )
6 df-nfc 2167 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
7 df-nfc 2167 . 2  |-  ( F/_ x B  <->  A. y F/ x  y  e.  B )
85, 6, 73bitr4g 212 1  |-  ( ph  ->  ( F/_ x A  <->  F/_ x B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98   A.wal 1241    = wceq 1243   F/wnf 1349    e. wcel 1393   F/_wnfc 2165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167
This theorem is referenced by:  nfopd  3566  dfnfc2  3598  nfimad  4677  nffvd  5187
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