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Theorem cbv1v 1747
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 16-Jun-2019.)
Hypotheses
Ref Expression
cbv1v.1  |-  F/ x ph
cbv1v.2  |-  F/ y
ph
cbv1v.3  |-  ( ph  ->  F/ y ps )
cbv1v.4  |-  ( ph  ->  F/ x ch )
cbv1v.5  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
Assertion
Ref Expression
cbv1v  |-  ( ph  ->  ( A. x ps 
->  A. y ch )
)
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    ch( x, y)

Proof of Theorem cbv1v
StepHypRef Expression
1 cbv1v.2 . . . . 5  |-  F/ y
ph
2 cbv1v.3 . . . . 5  |-  ( ph  ->  F/ y ps )
31, 2nfim1 1571 . . . 4  |-  F/ y ( ph  ->  ps )
4 cbv1v.1 . . . . 5  |-  F/ x ph
5 cbv1v.4 . . . . 5  |-  ( ph  ->  F/ x ch )
64, 5nfim1 1571 . . . 4  |-  F/ x
( ph  ->  ch )
7 cbv1v.5 . . . . . 6  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
87com12 30 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  ( ps  ->  ch ) ) )
98a2d 26 . . . 4  |-  ( x  =  y  ->  (
( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
103, 6, 9cbv3v 1744 . . 3  |-  ( A. x ( ph  ->  ps )  ->  A. y
( ph  ->  ch )
)
11419.21 1583 . . 3  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
12119.21 1583 . . 3  |-  ( A. y ( ph  ->  ch )  <->  ( ph  ->  A. y ch ) )
1310, 11, 123imtr3i 200 . 2  |-  ( (
ph  ->  A. x ps )  ->  ( ph  ->  A. y ch ) )
1413pm2.86i 99 1  |-  ( ph  ->  ( A. x ps 
->  A. y ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1351   F/wnf 1460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461
This theorem is referenced by:  cbv2w  1750
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