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Theorem cbvalh 1683
Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
cbvalh.1  |-  ( ph  ->  A. y ph )
cbvalh.2  |-  ( ps 
->  A. x ps )
cbvalh.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvalh  |-  ( A. x ph  <->  A. y ps )

Proof of Theorem cbvalh
StepHypRef Expression
1 cbvalh.1 . . 3  |-  ( ph  ->  A. y ph )
2 cbvalh.2 . . 3  |-  ( ps 
->  A. x ps )
3 cbvalh.3 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
43biimpd 142 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
51, 2, 4cbv3h 1678 . 2  |-  ( A. x ph  ->  A. y ps )
63equcoms 1641 . . . 4  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
76biimprd 156 . . 3  |-  ( y  =  x  ->  ( ps  ->  ph ) )
82, 1, 7cbv3h 1678 . 2  |-  ( A. y ps  ->  A. x ph )
95, 8impbii 124 1  |-  ( A. x ph  <->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  cbval  1684  sb8h  1782  cbvalv  1842  sb9v  1902  sb8euh  1971
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