Theorem sbcthdv 2923

Description: Deduction version of sbcth 2922. (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

Step	Hyp	Ref	Expression
1		sbcthdv.1	. . 3 |- ( ph -> ps )
2	1	alrimiv 1846	. 2 |- ( ph -> A. x ps )
3		spsbc 2920	. 2 |- ( A e. V -> ( A. x ps -> [. A / x ]. ps ) )
4	2, 3	mpan9 279	1 |- ( ( ph /\ A e. V ) -> [. A / x ]. ps )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1329   e. wcel 1480   [.wsbc 2909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688  df-sbc 2910

This theorem is referenced by: (None)