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Theorem nfbidf 1790
Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
Hypotheses
Ref Expression
nfbidf.1  |-  F/ x ph
nfbidf.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
nfbidf  |-  ( ph  ->  ( F/ x ps  <->  F/ x ch ) )

Proof of Theorem nfbidf
StepHypRef Expression
1 nfbidf.1 . . 3  |-  F/ x ph
2 nfbidf.2 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2albid 1788 . . . 4  |-  ( ph  ->  ( A. x ps  <->  A. x ch ) )
42, 3imbi12d 312 . . 3  |-  ( ph  ->  ( ( ps  ->  A. x ps )  <->  ( ch  ->  A. x ch )
) )
51, 4albid 1788 . 2  |-  ( ph  ->  ( A. x ( ps  ->  A. x ps )  <->  A. x ( ch 
->  A. x ch )
) )
6 df-nf 1554 . 2  |-  ( F/ x ps  <->  A. x
( ps  ->  A. x ps ) )
7 df-nf 1554 . 2  |-  ( F/ x ch  <->  A. x
( ch  ->  A. x ch ) )
85, 6, 73bitr4g 280 1  |-  ( ph  ->  ( F/ x ps  <->  F/ x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   F/wnf 1553
This theorem is referenced by:  drnf2  2058  dvelimdf  2066  nfsb4tOLD  2155  dvelimdfOLD  2157  nfcjust  2559  nfceqdf  2570  nfsb4twAUX7  29477  nfsb4tOLD7  29646  nfsb4tw2AUXOLD7  29647  dvelimdfOLD7  29652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551  df-nf 1554
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