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Theorem sbcimdv 2996
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 2013 . . . 4  |-  ( ph  ->  A. x ( ps 
->  ch ) )
3 a4sbc 2947 . . . 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 2976 . . 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 1532    e. wcel 1621   [.wsbc 2935
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-sbc 2936
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