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Theorem sbcimdv 2904
Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1391). (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 2848 . 2  |-  ( [. A  /  x ]. ps  ->  A  e.  _V )
2 sbcimdv.1 . . . . 5  |-  ( ph  ->  ( ps  ->  ch ) )
32alrimiv 1802 . . . 4  |-  ( ph  ->  A. x ( ps 
->  ch ) )
4 spsbc 2851 . . . 4  |-  ( A  e.  _V  ->  ( A. x ( ps  ->  ch )  ->  [. A  /  x ]. ( ps  ->  ch ) ) )
5 sbcim1 2887 . . . 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 80 . 2  |-  ( ph  ->  ( [. A  /  x ]. ps  ->  ( A  e.  _V  ->  [. A  /  x ]. ch ) ) )
81, 7mpdi 42 1  |-  ( ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1287    e. wcel 1438   _Vcvv 2619   [.wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841
This theorem is referenced by: (None)
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