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Theorem sbcthdv 3019
Description: Deduction version of sbcth 3018. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
sbcthdv.1  |-  ( ph  ->  ps )
Assertion
Ref Expression
sbcthdv  |-  ( (
ph  /\  A  e.  V )  ->  [. A  /  x ]. ps )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    A( x)    V( x)

Proof of Theorem sbcthdv
StepHypRef Expression
1 sbcthdv.1 . . 3  |-  ( ph  ->  ps )
21alrimiv 1621 . 2  |-  ( ph  ->  A. x ps )
3 spsbc 3016 . 2  |-  ( A  e.  V  ->  ( A. x ps  ->  [. A  /  x ]. ps )
)
42, 3mpan9 455 1  |-  ( (
ph  /\  A  e.  V )  ->  [. A  /  x ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530    e. wcel 1696   [.wsbc 3004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803  df-sbc 3005
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