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Theorem cbv2 1980
Description: Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (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 cbv2.2 . . . 4  |-  F/ y
ph
21nfri 1778 . . 3  |-  ( ph  ->  A. y ph )
3 cbv2.1 . . . . 5  |-  F/ x ph
43nfal 1864 . . . 4  |-  F/ x A. y ph
54nfri 1778 . . 3  |-  ( A. y ph  ->  A. x A. y ph )
62, 5syl 16 . 2  |-  ( ph  ->  A. x A. y ph )
7 cbv2.3 . . . 4  |-  ( ph  ->  F/ y ps )
87nfrd 1779 . . 3  |-  ( ph  ->  ( ps  ->  A. y ps ) )
9 cbv2.4 . . . 4  |-  ( ph  ->  F/ x ch )
109nfrd 1779 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
11 cbv2.5 . . 3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
128, 10, 11cbv2h 1979 . 2  |-  ( A. x A. y ph  ->  ( A. x ps  <->  A. y ch ) )
136, 12syl 16 1  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   F/wnf 1553
This theorem is referenced by:  cbvald  1986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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