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Theorem cbv2 2113
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style. (Revised by Wolf Lammen, 13-May-2018.)
Hypotheses
Ref Expression
cbv2.1  |-  F/ x ph
cbv2.2  |-  F/ y
ph
cbv2.3  |-  ( ph  ->  F/ y ps )
cbv2.4  |-  ( ph  ->  F/ x ch )
cbv2.5  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
cbv2  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )

Proof of Theorem cbv2
StepHypRef Expression
1 cbv2.1 . . 3  |-  F/ x ph
2 cbv2.2 . . . 4  |-  F/ y
ph
32nfri 1952 . . 3  |-  ( ph  ->  A. y ph )
41, 3alrimi 1955 . 2  |-  ( ph  ->  A. x A. y ph )
5 cbv2.3 . . . 4  |-  ( ph  ->  F/ y ps )
65nfrd 1953 . . 3  |-  ( ph  ->  ( ps  ->  A. y ps ) )
7 cbv2.4 . . . 4  |-  ( ph  ->  F/ x ch )
87nfrd 1953 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
9 cbv2.5 . . 3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
106, 8, 9cbv2h 2112 . 2  |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch ) )
114, 10syl 17 1  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1442   F/wnf 1667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668
This theorem is referenced by:  cbvald  2118  sb9  2255  wl-cbvalnaed  31865  wl-sb8t  31880
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