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Theorem sbcthdv 3176
Description: Deduction version of sbcth 3175. (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 1641 . 2  |-  ( ph  ->  A. x ps )
3 spsbc 3173 . 2  |-  ( A  e.  V  ->  ( A. x ps  ->  [. A  /  x ]. ps )
)
42, 3mpan9 456 1  |-  ( (
ph  /\  A  e.  V )  ->  [. A  /  x ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549    e. wcel 1725   [.wsbc 3161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-v 2958  df-sbc 3162
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