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Theorem nfceqdf 2493
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 2425 . . . 4  |-  ( ph  ->  ( y  e.  A  <->  y  e.  B ) )
41, 3nfbidf 1775 . . 3  |-  ( ph  ->  ( F/ x  y  e.  A  <->  F/ x  y  e.  B )
)
54albidv 1625 . 2  |-  ( ph  ->  ( A. y F/ x  y  e.  A  <->  A. y F/ x  y  e.  B ) )
6 df-nfc 2483 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
7 df-nfc 2483 . 2  |-  ( F/_ x B  <->  A. y F/ x  y  e.  B )
85, 6, 73bitr4g 279 1  |-  ( ph  ->  ( F/_ x A  <->  F/_ x B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1540   F/wnf 1544    = wceq 1642    e. wcel 1710   F/_wnfc 2481
This theorem is referenced by:  nfopd  3894  dfnfc2  3926  nfimad  5103  nffvd  5617  riotasvdOLD  6435  riotasv2d  6436  riotasv2dOLD  6437  riotasv3dOLD  6441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2351  df-clel 2354  df-nfc 2483
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