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Theorem nfceqdf 2418
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 2350 . . . 4  |-  ( ph  ->  ( y  e.  A  <->  y  e.  B ) )
41, 3nfbidf 1754 . . 3  |-  ( ph  ->  ( F/ x  y  e.  A  <->  F/ x  y  e.  B )
)
54albidv 1611 . 2  |-  ( ph  ->  ( A. y F/ x  y  e.  A  <->  A. y F/ x  y  e.  B ) )
6 df-nfc 2408 . 2  |-  ( F/_ x A  <->  A. y F/ x  y  e.  A )
7 df-nfc 2408 . 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 1527   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406
This theorem is referenced by:  nfopd  3813  dfnfc2  3845  nfimad  5021  nffvd  5534  riotasvdOLD  6348  riotasv2d  6349  riotasv2dOLD  6350  riotasv3dOLD  6354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-cleq 2276  df-clel 2279  df-nfc 2408
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