Theorem rcla4e 1875

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-17 973	. . . 4 |- (y e. A -> A.x y e. A)
2		ax-17 973	. . . . 5 |- (A e. B -> A.x A e. B)
3		rcla4.1	. . . . 5 |- (ps -> A.xps)
4	2, 3	hban 1011	. . . 4 |- ((A e. B /\ ps) -> A.x(A e. B /\ ps))
5		eleq1 1537	. . . . 5 |- (x = A -> (x e. B <-> A e. B))
6		rcla4.2	. . . . 5 |- (x = A -> (ph <-> ps))
7	5, 6	anbi12d 630	. . . 4 |- (x = A -> ((x e. B /\ ph) <-> (A e. B /\ ps)))
8	1, 4, 7	cla4egf 1864	. . 3 |- (A e. B -> ((A e. B /\ ps) -> E.x(x e. B /\ ph)))
9	8	anabsi5 497	. 2 |- ((A e. B /\ ps) -> E.x(x e. B /\ ph))
10		df-rex 1653	. 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 956   = wceq 958   e. wcel 960  E.wex 982  E.wrex 1649

This theorem is referenced by:  rcla4ev 1880  infcvgaux1 7219  fgsb 10555  fgsb2 10560

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rex 1653  df-v 1815

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