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Theorem nfceqdf 2573
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 2505 . . . 4  |-  ( ph  ->  ( y  e.  A  <->  y  e.  B ) )
41, 3nfbidf 1791 . . 3  |-  ( ph  ->  ( F/ x  y  e.  A  <->  F/ x  y  e.  B )
)
54albidv 1636 . 2  |-  ( ph  ->  ( A. y F/ x  y  e.  A  <->  A. y F/ x  y  e.  B ) )
6 df-nfc 2563 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
7 df-nfc 2563 . 2  |-  ( F/_ x B  <->  A. y F/ x  y  e.  B )
85, 6, 73bitr4g 281 1  |-  ( ph  ->  ( F/_ x A  <->  F/_ x B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550   F/wnf 1554    = wceq 1653    e. wcel 1726   F/_wnfc 2561
This theorem is referenced by:  nfopd  4003  dfnfc2  4035  nfimad  5215  nffvd  5740  riotasvdOLD  6596  riotasv2d  6597  riotasv2dOLD  6598  riotasv3dOLD  6602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-cleq 2431  df-clel 2434  df-nfc 2563
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