Theorem sbcthdv 3451

Description: Deduction version of sbcth 3450. (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 1855	. 2 |- ( ph -> A. x ps )
3		spsbc 3448	. 2 |- ( A e. V -> ( A. x ps -> [. A / x ]. ps ) )
4	2, 3	mpan9 486	1 |- ( ( ph /\ A e. V ) -> [. A / x ]. ps )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   e. wcel 1990   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by: (None)