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Theorem sbcimdv 3395
Description: Substitution analog of Theorem 19.20 of [Margaris] p. 90 (alim 1633). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.)
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 sbcimdv.1 . . . 4  |-  ( ph  ->  ( ps  ->  ch ) )
21alrimiv 1720 . . 3  |-  ( ph  ->  A. x ( ps 
->  ch ) )
3 spsbc 3340 . . 3  |-  ( A  e.  _V  ->  ( A. x ( ps  ->  ch )  ->  [. A  /  x ]. ( ps  ->  ch ) ) )
4 sbcim1 3376 . . 3  |-  ( [. A  /  x ]. ( ps  ->  ch )  -> 
( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )
52, 3, 4syl56 34 . 2  |-  ( A  e.  _V  ->  ( ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) )
6 sbcex 3337 . . . . 5  |-  ( [. A  /  x ]. ps  ->  A  e.  _V )
76con3i 135 . . . 4  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. ps )
87pm2.21d 106 . . 3  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )
98a1d 25 . 2  |-  ( -.  A  e.  _V  ->  (
ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) )
105, 9pm2.61i 164 1  |-  ( ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1393    e. wcel 1819   _Vcvv 3109   [.wsbc 3327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111  df-sbc 3328
This theorem is referenced by:  esum2dlem  28264
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