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Theorem sbcimdv 3209
Description: Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)
Hypothesis
Ref Expression
sbcimdv.1  |-  ( ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
sbcimdv  |-  ( (
ph  /\  A  e.  V )  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)    V( x)

Proof of Theorem sbcimdv
StepHypRef Expression
1 sbcimdv.1 . . . . 5  |-  ( ph  ->  ( ps  ->  ch ) )
21alrimiv 1641 . . . 4  |-  ( ph  ->  A. x ( ps 
->  ch ) )
3 spsbc 3160 . . . 4  |-  ( A  e.  V  ->  ( A. x ( ps  ->  ch )  ->  [. A  /  x ]. ( ps  ->  ch ) ) )
42, 3syl5 30 . . 3  |-  ( A  e.  V  ->  ( ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) )
5 sbcimg 3189 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ps  ->  ch ) 
<->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
) )
64, 5sylibd 206 . 2  |-  ( A  e.  V  ->  ( ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) )
76impcom 420 1  |-  ( (
ph  /\  A  e.  V )  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549    e. wcel 1725   [.wsbc 3148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-v 2945  df-sbc 3149
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