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Theorem sbcimdv 2944
Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1416). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
Hypothesis
Ref Expression
sbcimdv.1  |-  ( ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
sbcimdv  |-  ( ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
)
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)

Proof of Theorem sbcimdv
StepHypRef Expression
1 sbcex 2888 . 2  |-  ( [. A  /  x ]. ps  ->  A  e.  _V )
2 sbcimdv.1 . . . . 5  |-  ( ph  ->  ( ps  ->  ch ) )
32alrimiv 1828 . . . 4  |-  ( ph  ->  A. x ( ps 
->  ch ) )
4 spsbc 2891 . . . 4  |-  ( A  e.  _V  ->  ( A. x ( ps  ->  ch )  ->  [. A  /  x ]. ( ps  ->  ch ) ) )
5 sbcim1 2927 . . . 4  |-  ( [. A  /  x ]. ( ps  ->  ch )  -> 
( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )
63, 4, 5syl56 34 . . 3  |-  ( A  e.  _V  ->  ( ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) )
76com3l 81 . 2  |-  ( ph  ->  ( [. A  /  x ]. ps  ->  ( A  e.  _V  ->  [. A  /  x ]. ch ) ) )
81, 7mpdi 43 1  |-  ( ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1312    e. wcel 1463   _Vcvv 2658   [.wsbc 2880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sbc 2881
This theorem is referenced by: (None)
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