Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-cbv3tb Structured version   Visualization version   Unicode version

Theorem bj-cbv3tb 31312
Description: Closed form of cbv3 2108. (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 1969 . . . 4  |-  ( F/ x ps  ->  ( E. x ps  <->  ps )
)
21biimpd 211 . . 3  |-  ( F/ x ps  ->  ( E. x ps  ->  ps ) )
32alimi 1684 . 2  |-  ( A. y F/ x ps  ->  A. y ( E. x ps  ->  ps ) )
4 nfr 1951 . . 3  |-  ( F/ y ph  ->  ( ph  ->  A. y ph )
)
54alimi 1684 . 2  |-  ( A. x F/ y ph  ->  A. x ( ph  ->  A. y ph ) )
6 bj-cbv3ta 31311 . 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 662 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 371   A.wal 1442   E.wex 1663   F/wnf 1667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator