Theorem cla4gft 16406

Description: A closed version of cla4gf.

Hypotheses

Ref	Expression
cla4gft.1	|- (ps -> A.xps)
cla4gft.2	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
cla4gft	|- (A.x(x = A -> (ph <-> ps)) -> (A e. B -> (A.xph -> ps)))

Distinct variable groups:   x,y   y,A

Proof of Theorem cla4gft

Step	Hyp	Ref	Expression
1		hba1 1350	. . . . 5 |- (A.x(x = A -> (ph <-> ps)) -> A.xA.x(x = A -> (ph <-> ps)))
2		bi1 165	. . . . . . 7 |- ((ph <-> ps) -> (ph -> ps))
3	2	imim2i 11	. . . . . 6 |- ((x = A -> (ph <-> ps)) -> (x = A -> (ph -> ps)))
4	3	a4s 1330	. . . . 5 |- (A.x(x = A -> (ph <-> ps)) -> (x = A -> (ph -> ps)))
5	1, 4	eximd 1410	. . . 4 |- (A.x(x = A -> (ph <-> ps)) -> (E.x x = A -> E.x(ph -> ps)))
6		cla4gft.2	. . . . 5 |- (y e. A -> A.x y e. A)
7		ax-17 1317	. . . . 5 |- (x = A -> A.y x = A)
8	6, 7	isseta 16405	. . . 4 |- (A e. _V <-> E.x x = A)
9	5, 8	syl5ib 223	. . 3 |- (A.x(x = A -> (ph <-> ps)) -> (A e. _V -> E.x(ph -> ps)))
10		cla4gft.1	. . . 4 |- (ps -> A.xps)
11	10	19.36 1429	. . 3 |- (E.x(ph -> ps) <-> (A.xph -> ps))
12	9, 11	syl6ib 229	. 2 |- (A.x(x = A -> (ph <-> ps)) -> (A e. _V -> (A.xph -> ps)))
13		elisset 2299	. 2 |- (A e. B -> A e. _V)
14	12, 13	syl5 20	1 |- (A.x(x = A -> (ph <-> ps)) -> (A e. B -> (A.xph -> ps)))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292

This theorem is referenced by:  rcla4t 16407

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-v 2294

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