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Theorem sbcimdv 2946
Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1418). (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 2890 . 2  |-  ( [. A  /  x ]. ps  ->  A  e.  _V )
2 sbcimdv.1 . . . . 5  |-  ( ph  ->  ( ps  ->  ch ) )
32alrimiv 1830 . . . 4  |-  ( ph  ->  A. x ( ps 
->  ch ) )
4 spsbc 2893 . . . 4  |-  ( A  e.  _V  ->  ( A. x ( ps  ->  ch )  ->  [. A  /  x ]. ( ps  ->  ch ) ) )
5 sbcim1 2929 . . . 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 1314    e. wcel 1465   _Vcvv 2660   [.wsbc 2882
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883
This theorem is referenced by: (None)
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