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Theorem bj-cbv2hv 30858
Description: Version of cbv2h 2046 with a dv condition, which does not require ax-13 2026. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbv2hv.1  |-  ( ph  ->  ( ps  ->  A. y ps ) )
bj-cbv2hv.2  |-  ( ph  ->  ( ch  ->  A. x ch ) )
bj-cbv2hv.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
bj-cbv2hv  |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    ch( x, y)

Proof of Theorem bj-cbv2hv
StepHypRef Expression
1 bj-cbv2hv.1 . . 3  |-  ( ph  ->  ( ps  ->  A. y ps ) )
2 bj-cbv2hv.2 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
3 bj-cbv2hv.3 . . . 4  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
4 biimp 193 . . . 4  |-  ( ( ps  <->  ch )  ->  ( ps  ->  ch ) )
53, 4syl6 31 . . 3  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
61, 2, 5bj-cbv1hv 30857 . 2  |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch ) )
7 equcomi 1817 . . . . 5  |-  ( y  =  x  ->  x  =  y )
8 biimpr 198 . . . . 5  |-  ( ( ps  <->  ch )  ->  ( ch  ->  ps ) )
97, 3, 8syl56 32 . . . 4  |-  ( ph  ->  ( y  =  x  ->  ( ch  ->  ps ) ) )
102, 1, 9bj-cbv1hv 30857 . . 3  |-  ( A. y A. x ph  ->  ( A. y ch  ->  A. x ps ) )
1110alcoms 1867 . 2  |-  ( A. x A. y ph  ->  ( A. y ch  ->  A. x ps ) )
126, 11impbid 190 1  |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878
This theorem depends on definitions:  df-bi 185  df-ex 1634  df-nf 1638
This theorem is referenced by:  bj-cbv2v  30859
  Copyright terms: Public domain W3C validator