Theorem rcla4t 16407

Description: A closed version of rcla4.

Hypothesis

Ref	Expression
rcla4t.1	|- (ps -> A.xps)

Assertion

Ref	Expression
rcla4t	|- (A.x(x = A -> (ph <-> ps)) -> (A e. B -> (A.x e. B ph -> ps)))

Distinct variable groups:   x,A   x,B

Proof of Theorem rcla4t

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . . . . . . 10 |- (x = A -> (x e. B <-> A e. B))
2	1	adantr 425	. . . . . . . . 9 |- ((x = A /\ (ph <-> ps)) -> (x e. B <-> A e. B))
3		simpr 350	. . . . . . . . 9 |- ((x = A /\ (ph <-> ps)) -> (ph <-> ps))
4	2, 3	imbi12d 688	. . . . . . . 8 |- ((x = A /\ (ph <-> ps)) -> ((x e. B -> ph) <-> (A e. B -> ps)))
5	4	ex 402	. . . . . . 7 |- (x = A -> ((ph <-> ps) -> ((x e. B -> ph) <-> (A e. B -> ps))))
6	5	a2i 10	. . . . . 6 |- ((x = A -> (ph <-> ps)) -> (x = A -> ((x e. B -> ph) <-> (A e. B -> ps))))
7	6	alimi 1338	. . . . 5 |- (A.x(x = A -> (ph <-> ps)) -> A.x(x = A -> ((x e. B -> ph) <-> (A e. B -> ps))))
8		ax-17 1317	. . . . . . 7 |- (A e. B -> A.x A e. B)
9		rcla4t.1	. . . . . . 7 |- (ps -> A.xps)
10	8, 9	hbim 1354	. . . . . 6 |- ((A e. B -> ps) -> A.x(A e. B -> ps))
11		ax-17 1317	. . . . . 6 |- (z e. A -> A.x z e. A)
12	10, 11	cla4gft 16406	. . . . 5 |- (A.x(x = A -> ((x e. B -> ph) <-> (A e. B -> ps))) -> (A e. B -> (A.x(x e. B -> ph) -> (A e. B -> ps))))
13	7, 12	syl 12	. . . 4 |- (A.x(x = A -> (ph <-> ps)) -> (A e. B -> (A.x(x e. B -> ph) -> (A e. B -> ps))))
14		df-ral 2109	. . . 4 |- (A.x e. B ph <-> A.x(x e. B -> ph))
15	13, 14	syl7ib 233	. . 3 |- (A.x(x = A -> (ph <-> ps)) -> (A e. B -> (A.x e. B ph -> (A e. B -> ps))))
16	15	com34 40	. 2 |- (A.x(x = A -> (ph <-> ps)) -> (A e. B -> (A e. B -> (A.x e. B ph -> ps))))
17	16	pm2.43d 79	1 |- (A.x(x = A -> (ph <-> ps)) -> (A e. B -> (A.x e. B ph -> ps)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105

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-v 2294

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