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Theorem cbv2h 2123
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 11-May-1993.)
Hypotheses
Ref Expression
cbv2h.1  |-  ( ph  ->  ( ps  ->  A. y ps ) )
cbv2h.2  |-  ( ph  ->  ( ch  ->  A. x ch ) )
cbv2h.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
cbv2h  |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch ) )

Proof of Theorem cbv2h
StepHypRef Expression
1 cbv2h.1 . . 3  |-  ( ph  ->  ( ps  ->  A. y ps ) )
2 cbv2h.2 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
3 cbv2h.3 . . . 4  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
4 biimp 198 . . . 4  |-  ( ( ps  <->  ch )  ->  ( ps  ->  ch ) )
53, 4syl6 34 . . 3  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
61, 2, 5cbv1h 2122 . 2  |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch ) )
7 equcomi 1872 . . . . 5  |-  ( y  =  x  ->  x  =  y )
8 biimpr 203 . . . . 5  |-  ( ( ps  <->  ch )  ->  ( ch  ->  ps ) )
97, 3, 8syl56 35 . . . 4  |-  ( ph  ->  ( y  =  x  ->  ( ch  ->  ps ) ) )
102, 1, 9cbv1h 2122 . . 3  |-  ( A. y A. x ph  ->  ( A. y ch  ->  A. x ps ) )
1110alcoms 1932 . 2  |-  ( A. x A. y ph  ->  ( A. y ch  ->  A. x ps ) )
126, 11impbid 195 1  |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679
This theorem is referenced by:  cbv2  2124  eujustALT  2313
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