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Theorem cbv2 1708
Description: Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
Hypotheses
Ref Expression
cbv2.1  |-  F/ x ph
cbv2.2  |-  F/ y
ph
cbv2.3  |-  ( ph  ->  F/ y ps )
cbv2.4  |-  ( ph  ->  F/ x ch )
cbv2.5  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
cbv2  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )

Proof of Theorem cbv2
StepHypRef Expression
1 cbv2.2 . . . 4  |-  F/ y
ph
21nfri 1482 . . 3  |-  ( ph  ->  A. y ph )
3 cbv2.1 . . . . 5  |-  F/ x ph
43nfal 1538 . . . 4  |-  F/ x A. y ph
54nfri 1482 . . 3  |-  ( A. y ph  ->  A. x A. y ph )
62, 5syl 14 . 2  |-  ( ph  ->  A. x A. y ph )
7 cbv2.3 . . . 4  |-  ( ph  ->  F/ y ps )
87nfrd 1483 . . 3  |-  ( ph  ->  ( ps  ->  A. y ps ) )
9 cbv2.4 . . . 4  |-  ( ph  ->  F/ x ch )
109nfrd 1483 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
11 cbv2.5 . . 3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
128, 10, 11cbv2h 1707 . 2  |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch ) )
136, 12syl 14 1  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1312   F/wnf 1419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420
This theorem is referenced by:  cbvald  1875  cbvrald  12797
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