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Theorem bj-cbv3tb 33754
Description: Closed form of cbv3 1984. (Contributed by BJ, 2-May-2019.)
Assertion
Ref Expression
bj-cbv3tb  |-  ( A. x A. y ( x  =  y  ->  ( ph  ->  ps ) )  ->  ( ( A. y F/ x ps  /\  A. x F/ y ph )  ->  ( A. x ph  ->  A. y ps )
) )

Proof of Theorem bj-cbv3tb
StepHypRef Expression
1 19.9t 1838 . . . 4  |-  ( F/ x ps  ->  ( E. x ps  <->  ps )
)
21biimpd 207 . . 3  |-  ( F/ x ps  ->  ( E. x ps  ->  ps ) )
32alimi 1614 . 2  |-  ( A. y F/ x ps  ->  A. y ( E. x ps  ->  ps ) )
4 nfr 1821 . . 3  |-  ( F/ y ph  ->  ( ph  ->  A. y ph )
)
54alimi 1614 . 2  |-  ( A. x F/ y ph  ->  A. x ( ph  ->  A. y ph ) )
6 bj-cbv3ta 33753 . 2  |-  ( A. x A. y ( x  =  y  ->  ( ph  ->  ps ) )  ->  ( ( A. y ( E. x ps  ->  ps )  /\  A. x ( ph  ->  A. y ph ) )  ->  ( A. x ph  ->  A. y ps )
) )
73, 5, 6syl2ani 656 1  |-  ( A. x A. y ( x  =  y  ->  ( ph  ->  ps ) )  ->  ( ( A. y F/ x ps  /\  A. x F/ y ph )  ->  ( A. x ph  ->  A. y ps )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1377   E.wex 1596   F/wnf 1599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600
This theorem is referenced by: (None)
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