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Theorem sbcimdv 3054
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 1618 . . . 4  |-  ( ph  ->  A. x ( ps 
->  ch ) )
3 spsbc 3005 . . . 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 3034 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ps  ->  ch ) 
<->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
) )
64, 5sylibd 207 . 2  |-  ( A  e.  V  ->  ( ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) )
76impcom 421 1  |-  ( (
ph  /\  A  e.  V )  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   A.wal 1528    e. wcel 1685   [.wsbc 2993
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-sbc 2994
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