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Theorem cbv1 1932
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
cbv1.1  |-  ( ph  ->  F/ y ps )
cbv1.2  |-  ( ph  ->  F/ x ch )
cbv1.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
Assertion
Ref Expression
cbv1  |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch ) )

Proof of Theorem cbv1
StepHypRef Expression
1 cbv1.1 . . 3  |-  ( ph  ->  F/ y ps )
21nfrd 1755 . 2  |-  ( ph  ->  ( ps  ->  A. y ps ) )
3 cbv1.2 . . 3  |-  ( ph  ->  F/ x ch )
43nfrd 1755 . 2  |-  ( ph  ->  ( ch  ->  A. x ch ) )
5 cbv1.3 . 2  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
62, 4, 5cbv1h 1931 1  |-  ( A. x A. y ph  ->  ( A. x ps  ->  A. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530   F/wnf 1534
This theorem is referenced by:  cbv3  1935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535
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