Theorem rcla4e 1863

Description: Restricted existential specialization with implicit substitution.

Hypotheses

Ref	Expression
rcla4.1	|- (ps -> A.xps)
rcla4.2	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
rcla4e	|- ((A e. B /\ ps) -> E.x e. B ph)

Distinct variable groups:   x,A   x,B

Proof of Theorem rcla4e

Step	Hyp	Ref	Expression
1		ax-17 968	. . . 4 |- (y e. A -> A.x y e. A)
2		ax-17 968	. . . . 5 |- (A e. B -> A.x A e. B)
3		rcla4.1	. . . . 5 |- (ps -> A.xps)
4	2, 3	hban 1006	. . . 4 |- ((A e. B /\ ps) -> A.x(A e. B /\ ps))
5		eleq1 1526	. . . . 5 |- (x = A -> (x e. B <-> A e. B))
6		rcla4.2	. . . . 5 |- (x = A -> (ph <-> ps))
7	5, 6	anbi12d 626	. . . 4 |- (x = A -> ((x e. B /\ ph) <-> (A e. B /\ ps)))
8	1, 4, 7	cla4egf 1852	. . 3 |- (A e. B -> ((A e. B /\ ps) -> E.x(x e. B /\ ph)))
9	8	anabsi5 494	. 2 |- ((A e. B /\ ps) -> E.x(x e. B /\ ph))
10		df-rex 1642	. 2 |- (E.x e. B ph <-> E.x(x e. B /\ ph))
11	9, 10	sylibr 200	1 |- ((A e. B /\ ps) -> E.x e. B ph)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  E.wrex 1638

This theorem is referenced by:  rcla4ev 1868  infcvgaux1 7154  fgsb 10444  fgsb2 10449

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  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-rex 1642  df-v 1803

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