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Theorem sbcthdv 3006
Description: Deduction version of sbcth 3005. (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 1617 . 2  |-  ( ph  ->  A. x ps )
3 spsbc 3003 . 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 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-an 360  df-ex 1529  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790  df-sbc 2992
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