Theorem rcla4edv 2383

Description: Restricted existential specialization, using implicit substitition. (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 403	. . . . . . 7 |- (x = A -> (ph -> (ps <-> ch)))
3	2	pm5.74d 645	. . . . . 6 |- (x = A -> ((ph -> ps) <-> (ph -> ch)))
4	3	rcla4ev 2381	. . . . 5 |- ((A e. B /\ (ph -> ch)) -> E.x e. B (ph -> ps))
5		r19.37av 2233	. . . . 5 |- (E.x e. B (ph -> ps) -> (ph -> E.x e. B ps))
6	4, 5	syl 12	. . . 4 |- ((A e. B /\ (ph -> ch)) -> (ph -> E.x e. B ps))
7	6	ex 402	. . 3 |- (A e. B -> ((ph -> ch) -> (ph -> E.x e. B ps)))
8	7	pm2.86d 87	. 2 |- (A e. B -> (ph -> (ch -> E.x e. B ps)))
9	8	impcom 378	1 |- ((ph /\ A e. B) -> (ch -> E.x e. B ps))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106

This theorem is referenced by:  gcdcllem1 13718  prtoptop 14919

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294

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