Theorem sbcthdv 2829

Description: Deduction version of sbcth 2828. (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 1795	. 2 |- ( ph -> A. x ps )
3		spsbc 2826	. 2 |- ( A e. V -> ( A. x ps -> [. A / x ]. ps ) )
4	2, 3	mpan9 275	1 |- ( ( ph /\ A e. V ) -> [. A / x ]. ps )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   e. wcel 1433   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603  df-sbc 2816

This theorem is referenced by: (None)