Theorem sbcthdv 1937

Description: Deduction version of sbcth 1936.

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 1281	. . 3 |- (ph -> A.xps)
3	2	adantr 389	. 2 |- ((ph /\ A e. B) -> A.xps)
4		a4sbc 1935	. . 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 951   e. wcel 955  [wsbc 1166

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932

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