Theorem rcla4 1867

Description: Restricted specialization with implicit substitution.

Hypotheses

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

Assertion

Ref	Expression
rcla4	|- (A e. B -> (A.x e. B ph -> ps))

Distinct variable groups:   x,A   x,B

Proof of Theorem rcla4

Step	Hyp	Ref	Expression
1		df-ral 1646	. . 3 |- (A.x e. B ph <-> A.x(x e. B -> ph))
2		ax-17 969	. . . . 5 |- (y e. A -> A.x y e. A)
3		ax-17 969	. . . . . 6 |- (A e. B -> A.x A e. B)
4		rcla4.1	. . . . . 6 |- (ps -> A.xps)
5	3, 4	hbim 1005	. . . . 5 |- ((A e. B -> ps) -> A.x(A e. B -> ps))
6		eleq1 1531	. . . . . 6 |- (x = A -> (x e. B <-> A e. B))
7		rcla4.2	. . . . . 6 |- (x = A -> (ph <-> ps))
8	6, 7	imbi12d 625	. . . . 5 |- (x = A -> ((x e. B -> ph) <-> (A e. B -> ps)))
9	2, 5, 8	cla4gf 1856	. . . 4 |- (A e. B -> (A.x(x e. B -> ph) -> (A e. B -> ps)))
10	9	pm2.43b 67	. . 3 |- (A.x(x e. B -> ph) -> (A e. B -> ps))
11	1, 10	sylbi 199	. 2 |- (A.x e. B ph -> (A e. B -> ps))
12	11	com12 11	1 |- (A e. B -> (A.x e. B ph -> ps))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952   = wceq 954   e. wcel 956  A.wral 1642

This theorem is referenced by:  rcla4v 1869  lble 6002  cau3i 6859  irredt 10259

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-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808

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