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Theorem cbv3h 1672
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.)
Hypotheses
Ref Expression
cbv3h.1  |-  ( ph  ->  A. y ph )
cbv3h.2  |-  ( ps 
->  A. x ps )
cbv3h.3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
cbv3h  |-  ( A. x ph  ->  A. y ps )

Proof of Theorem cbv3h
StepHypRef Expression
1 cbv3h.1 . . 3  |-  ( ph  ->  A. y ph )
21nfi 1392 . 2  |-  F/ y
ph
3 cbv3h.2 . . 3  |-  ( ps 
->  A. x ps )
43nfi 1392 . 2  |-  F/ x ps
5 cbv3h.3 . 2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
62, 4, 5cbv3 1671 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   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-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  cbvalh  1677  ax16  1735  ax16i  1780  cleqh  2179
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