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Theorem cbvalh 1652
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 136 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
51, 2, 4cbv3h 1647 . 2  |-  ( A. x ph  ->  A. y ps )
63equcoms 1610 . . . 4  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
76biimprd 151 . . 3  |-  ( y  =  x  ->  ( ps  ->  ph ) )
82, 1, 7cbv3h 1647 . 2  |-  ( A. y ps  ->  A. x ph )
95, 8impbii 121 1  |-  ( A. x ph  <->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  cbval  1653  sb8h  1750  cbvalv  1810  sb9v  1870  sb8euh  1939
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