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Theorem bj-cbv3tb 31048
Description: Closed form of cbv3 2068. (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 1941 . . . 4  |-  ( F/ x ps  ->  ( E. x ps  <->  ps )
)
21biimpd 210 . . 3  |-  ( F/ x ps  ->  ( E. x ps  ->  ps ) )
32alimi 1680 . 2  |-  ( A. y F/ x ps  ->  A. y ( E. x ps  ->  ps ) )
4 nfr 1923 . . 3  |-  ( F/ y ph  ->  ( ph  ->  A. y ph )
)
54alimi 1680 . 2  |-  ( A. x F/ y ph  ->  A. x ( ph  ->  A. y ph ) )
6 bj-cbv3ta 31047 . 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 660 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 370   A.wal 1435   E.wex 1659   F/wnf 1663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by: (None)
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