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Theorem cbv1 2022
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
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 1924 . . . 4  |-  F/ y ( ph  ->  ps )
4 cbv1.1 . . . . 5  |-  F/ x ph
5 cbv1.4 . . . . 5  |-  ( ph  ->  F/ x ch )
64, 5nfim1 1924 . . . 4  |-  F/ x
( ph  ->  ch )
7 cbv1.5 . . . . . 6  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
87com12 31 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  ( ps  ->  ch ) ) )
98a2d 26 . . . 4  |-  ( x  =  y  ->  (
( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
103, 6, 9cbv3 2020 . . 3  |-  ( A. x ( ph  ->  ps )  ->  A. y
( ph  ->  ch )
)
11419.21 1910 . . 3  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
12119.21 1910 . . 3  |-  ( A. y ( ph  ->  ch )  <->  ( ph  ->  A. y ch ) )
1310, 11, 123imtr3i 265 . 2  |-  ( (
ph  ->  A. x ps )  ->  ( ph  ->  A. y ch ) )
1413pm2.86i 101 1  |-  ( ph  ->  ( A. x ps 
->  A. y ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1396   F/wnf 1621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622
This theorem is referenced by:  cbv1h  2023
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