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Theorem cbv2h 1675
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-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 bi1 116 . . . 4  |-  ( ( ps  <->  ch )  ->  ( ps  ->  ch ) )
53, 4syl6 33 . . 3  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
61, 2, 5cbv1h 1674 . 2  |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch ) )
7 equcomi 1633 . . . . 5  |-  ( y  =  x  ->  x  =  y )
8 bi2 128 . . . . 5  |-  ( ( ps  <->  ch )  ->  ( ch  ->  ps ) )
97, 3, 8syl56 34 . . . 4  |-  ( ph  ->  ( y  =  x  ->  ( ch  ->  ps ) ) )
102, 1, 9cbv1h 1674 . . 3  |-  ( A. y A. x ph  ->  ( A. y ch  ->  A. x ps ) )
1110a7s 1384 . 2  |-  ( A. x A. y ph  ->  ( A. y ch  ->  A. x ps ) )
126, 11impbid 127 1  |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  cbv2  1676
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