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Theorem cbv2 2021
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
cbv2.1  |-  ( ph  ->  F/ y ps )
cbv2.2  |-  ( ph  ->  F/ x ch )
cbv2.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
cbv2  |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch ) )

Proof of Theorem cbv2
StepHypRef Expression
1 cbv2.1 . . 3  |-  ( ph  ->  F/ y ps )
21nfrd 1771 . 2  |-  ( ph  ->  ( ps  ->  A. y ps ) )
3 cbv2.2 . . 3  |-  ( ph  ->  F/ x ch )
43nfrd 1771 . 2  |-  ( ph  ->  ( ch  ->  A. x ch ) )
5 cbv2.3 . 2  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
62, 4, 5cbv2h 2020 1  |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546   F/wnf 1550
This theorem is referenced by:  cbvald  2044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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