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Theorem sbcimdv 2888
Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1387). (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 2832 . 2  |-  ( [. A  /  x ]. ps  ->  A  e.  _V )
2 sbcimdv.1 . . . . 5  |-  ( ph  ->  ( ps  ->  ch ) )
32alrimiv 1797 . . . 4  |-  ( ph  ->  A. x ( ps 
->  ch ) )
4 spsbc 2835 . . . 4  |-  ( A  e.  _V  ->  ( A. x ( ps  ->  ch )  ->  [. A  /  x ]. ( ps  ->  ch ) ) )
5 sbcim1 2871 . . . 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 1283    e. wcel 1434   _Vcvv 2610   [.wsbc 2824
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825
This theorem is referenced by: (None)
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