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Theorem sbcimdv 3052
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 1617 . . . 4  |-  ( ph  ->  A. x ( ps 
->  ch ) )
3 spsbc 3003 . . . 4  |-  ( A  e.  V  ->  ( A. x ( ps  ->  ch )  ->  [. A  /  x ]. ( ps  ->  ch ) ) )
42, 3syl5 28 . . 3  |-  ( A  e.  V  ->  ( ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) )
5 sbcimg 3032 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ps  ->  ch ) 
<->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
) )
64, 5sylibd 205 . 2  |-  ( A  e.  V  ->  ( ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) )
76impcom 419 1  |-  ( (
ph  /\  A  e.  V )  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    e. wcel 1684   [.wsbc 2991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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