Theorem sbcthdvOLD 2460

Description: Deduction version of sbcth 2458.

Hypothesis

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

Assertion

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

Distinct variable group:   ph,x

Proof of Theorem sbcthdvOLD

Step	Hyp	Ref	Expression
1		sbcthdv.1	. . . 4 |- (ph -> ps)
2	1	19.21aiv 1664	. . 3 |- (ph -> A.xps)
3	2	adantr 425	. 2 |- ((ph /\ A e. B) -> A.xps)
4		a4sbc 2457	. . 3 |- (A e. B -> (A.xps -> [A / x]ps))
5	4	adantl 424	. 2 |- ((ph /\ A e. B) -> (A.xps -> [A / x]ps))
6	3, 5	mpd 29	1 |- ((ph /\ A e. B) -> [A / x]ps)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   e. wcel 1300  [wsbc 1534

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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