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Theorem cbv1 1973
Description: Rule used to change bound variables, using implicit substitution. Revised to format 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
cbv1.1  |-  F/ x ph
cbv1.2  |-  F/ y
ph
cbv1.3  |-  ( ph  ->  F/ y ps )
cbv1.4  |-  ( ph  ->  F/ x ch )
cbv1.5  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
Assertion
Ref Expression
cbv1  |-  ( ph  ->  ( A. x ps 
->  A. y ch )
)

Proof of Theorem cbv1
StepHypRef Expression
1 cbv1.2 . . . . 5  |-  F/ y
ph
2 cbv1.3 . . . . 5  |-  ( ph  ->  F/ y ps )
31, 2nfim1 1830 . . . 4  |-  F/ y ( ph  ->  ps )
4 cbv1.1 . . . . 5  |-  F/ x ph
5 cbv1.4 . . . . 5  |-  ( ph  ->  F/ x ch )
64, 5nfim1 1830 . . . 4  |-  F/ x
( ph  ->  ch )
7 cbv1.5 . . . . . 6  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
87com12 29 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  ( ps  ->  ch ) ) )
98a2d 24 . . . 4  |-  ( x  =  y  ->  (
( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
103, 6, 9cbv3 1971 . . 3  |-  ( A. x ( ph  ->  ps )  ->  A. y
( ph  ->  ch )
)
11419.21 1814 . . 3  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
12119.21 1814 . . 3  |-  ( A. y ( ph  ->  ch )  <->  ( ph  ->  A. y ch ) )
1310, 11, 123imtr3i 257 . 2  |-  ( (
ph  ->  A. x ps )  ->  ( ph  ->  A. y ch ) )
1413pm2.86i 94 1  |-  ( ph  ->  ( A. x ps 
->  A. y ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   F/wnf 1553
This theorem is referenced by:  cbv1h  1974
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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