Theorem rcla4edv 1875

Description: Restricted existential specialization with implicit substitution. (Contributed by FL, 17-Apr-2007.)

Hypothesis

Ref	Expression
rcla4dv.1	|- ((ph /\ x = A) -> (ps <-> ch))

Assertion

Ref	Expression
rcla4edv	|- ((ph /\ A e. B) -> (ch -> E.x e. B ps))

Distinct variable groups:   x,A   x,B   ph,x   ch,x

Proof of Theorem rcla4edv

Step	Hyp	Ref	Expression
1		rcla4dv.1	. . . . . . . 8 |- ((ph /\ x = A) -> (ps <-> ch))
2	1	expcom 374	. . . . . . 7 |- (x = A -> (ph -> (ps <-> ch)))
3	2	pm5.74d 584	. . . . . 6 |- (x = A -> ((ph -> ps) <-> (ph -> ch)))
4	3	rcla4ev 1873	. . . . 5 |- ((A e. B /\ (ph -> ch)) -> E.x e. B (ph -> ps))
5		r19.37av 1758	. . . . 5 |- (E.x e. B (ph -> ps) -> (ph -> E.x e. B ps))
6	4, 5	syl 10	. . . 4 |- ((A e. B /\ (ph -> ch)) -> (ph -> E.x e. B ps))
7	6	ex 373	. . 3 |- (A e. B -> ((ph -> ch) -> (ph -> E.x e. B ps)))
8	7	pm2.86d 71	. 2 |- (A e. B -> (ph -> (ch -> E.x e. B ps)))
9	8	impcom 351	1 |- ((ph /\ A e. B) -> (ch -> E.x e. B ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wrex 1643

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rex 1647  df-v 1808

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