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Theorem bj-nfbi 31217
Description: Closed form of nfbii 1689 (with df-bj-nf 31195 instead of df-nf 1662, which would require more axioms). (Contributed by BJ, 6-May-2019.)
Assertion
Ref Expression
bj-nfbi  |-  ( A. x ( ph  <->  ps )  ->  (FF/ x ph  <-> FF/ x ps ) )

Proof of Theorem bj-nfbi
StepHypRef Expression
1 exbi 1710 . . 3  |-  ( A. x ( ph  <->  ps )  ->  ( E. x ph  <->  E. x ps ) )
2 albi 1684 . . 3  |-  ( A. x ( ph  <->  ps )  ->  ( A. x ph  <->  A. x ps ) )
31, 2imbi12d 321 . 2  |-  ( A. x ( ph  <->  ps )  ->  ( ( E. x ph  ->  A. x ph )  <->  ( E. x ps  ->  A. x ps ) ) )
4 df-bj-nf 31195 . 2  |-  (FF/ x
ph 
<->  ( E. x ph  ->  A. x ph )
)
5 df-bj-nf 31195 . 2  |-  (FF/ x ps  <->  ( E. x ps  ->  A. x ps )
)
63, 4, 53bitr4g 291 1  |-  ( A. x ( ph  <->  ps )  ->  (FF/ x ph  <-> FF/ x ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435   E.wex 1657  FF/wnff 31194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676
This theorem depends on definitions:  df-bi 188  df-ex 1658  df-bj-nf 31195
This theorem is referenced by:  bj-nfxfr  31218
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