Theorem sbcthdv 2979

Description: Deduction version of sbcth 2978. (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 1874	. 2 |- ( ph -> A. x ps )
3		spsbc 2976	. 2 |- ( A e. V -> ( A. x ps -> [. A / x ]. ps ) )
4	2, 3	mpan9 281	1 |- ( ( ph /\ A e. V ) -> [. A / x ]. ps )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1351   e. wcel 2148   [.wsbc 2964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2741  df-sbc 2965

This theorem is referenced by: (None)