Theorem sbcthdv 2992

Description: Deduction version of sbcth 2991. (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 1885	. 2 |- ( ph -> A. x ps )
3		spsbc 2989	. 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 1362   e. wcel 2160   [.wsbc 2977

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754  df-sbc 2978

This theorem is referenced by: (None)