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Theorem cbv3h 2072
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 8-Jun-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 1670 . 2  |-  F/ y
ph
3 cbv3h.2 . . 3  |-  ( ps 
->  A. x ps )
43nfi 1670 . 2  |-  F/ x ps
5 cbv3h.3 . 2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
62, 4, 5cbv3 2071 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by:  cleqhOLD  2545
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