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Theorem cbv3 1672
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
Hypotheses
Ref Expression
cbv3.1  |-  F/ y
ph
cbv3.2  |-  F/ x ps
cbv3.3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
cbv3  |-  ( A. x ph  ->  A. y ps )

Proof of Theorem cbv3
StepHypRef Expression
1 cbv3.1 . . 3  |-  F/ y
ph
21nfal 1509 . 2  |-  F/ y A. x ph
3 cbv3.2 . . 3  |-  F/ x ps
4 cbv3.3 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
53, 4spim 1668 . 2  |-  ( A. x ph  ->  ps )
62, 5alrimi 1456 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283   F/wnf 1390
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-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  cbv3h  1673  cbv1  1674  mo2n  1971  mo23  1984  setindis  11047  bdsetindis  11049
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