Theorem sbcthdv 1944

Description: Deduction version of sbcth 1943.

Hypothesis

Ref	Expression
sbcthdv.1	|- (ph -> ps)

Assertion

Ref	Expression
sbcthdv	|- ((ph /\ A e. B) -> [A / x]ps)

Distinct variable group:   ph,x

Proof of Theorem sbcthdv

Step	Hyp	Ref	Expression
1		sbcthdv.1	. . . 4 |- (ph -> ps)
2	1	19.21aiv 1285	. . 3 |- (ph -> A.xps)
3	2	adantr 389	. 2 |- ((ph /\ A e. B) -> A.xps)
4		a4sbc 1942	. . 3 |- (A e. B -> (A.xps -> [A / x]ps))
5	4	adantl 388	. 2 |- ((ph /\ A e. B) -> (A.xps -> [A / x]ps))
6	3, 5	mpd 26	1 |- ((ph /\ A e. B) -> [A / x]ps)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 953   e. wcel 957  [wsbc 1169

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-sbc 1939

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