Theorem rgen2aOLD 2161

Description: Generalization rule for restricted quantification. Note that x and y needn't be distinct.

Hypothesis

Ref	Expression
rgen2a.1	|- ((x e. A /\ y e. A) -> ph)

Assertion

Ref	Expression
rgen2aOLD	|- A.x e. A A.y e. A ph

Distinct variable group:   y,A

Proof of Theorem rgen2aOLD

Step	Hyp	Ref	Expression
1		rgen2a.1	. . . . . . . 8 |- ((x e. A /\ y e. A) -> ph)
2	1	ex 402	. . . . . . 7 |- (x e. A -> (y e. A -> ph))
3	2	ax-gen 1305	. . . . . 6 |- A.y(x e. A -> (y e. A -> ph))
4		eleq1 1957	. . . . . . . . 9 |- (y = x -> (y e. A <-> x e. A))
5	4	a4s 1330	. . . . . . . 8 |- (A.y y = x -> (y e. A <-> x e. A))
6	5	imbi1d 675	. . . . . . 7 |- (A.y y = x -> ((y e. A -> (y e. A -> ph)) <-> (x e. A -> (y e. A -> ph))))
7	6	dral2 1516	. . . . . 6 |- (A.y y = x -> (A.y(y e. A -> (y e. A -> ph)) <-> A.y(x e. A -> (y e. A -> ph))))
8	3, 7	mpbiri 211	. . . . 5 |- (A.y y = x -> A.y(y e. A -> (y e. A -> ph)))
9		pm2.43 77	. . . . . 6 |- ((y e. A -> (y e. A -> ph)) -> (y e. A -> ph))
10	9	alimi 1338	. . . . 5 |- (A.y(y e. A -> (y e. A -> ph)) -> A.y(y e. A -> ph))
11		ax-1 4	. . . . 5 |- (A.y(y e. A -> ph) -> (x e. A -> A.y(y e. A -> ph)))
12	8, 10, 11	3syl 24	. . . 4 |- (A.y y = x -> (x e. A -> A.y(y e. A -> ph)))
13		ax-17 1317	. . . . . 6 |- (z e. A -> A.y z e. A)
14		eleq1 1957	. . . . . 6 |- (z = x -> (z e. A <-> x e. A))
15	13, 14	dvelim 1743	. . . . 5 |- (-. A.y y = x -> (x e. A -> A.y x e. A))
16	2	alimi 1338	. . . . 5 |- (A.y x e. A -> A.y(y e. A -> ph))
17	15, 16	syl6 25	. . . 4 |- (-. A.y y = x -> (x e. A -> A.y(y e. A -> ph)))
18	12, 17	pm2.61i 140	. . 3 |- (x e. A -> A.y(y e. A -> ph))
19		df-ral 2109	. . 3 |- (A.y e. A ph <-> A.y(y e. A -> ph))
20	18, 19	sylibr 217	. 2 |- (x e. A -> A.y e. A ph)
21	20	rgen 2159	1 |- A.x e. A A.y e. A ph

Syntax hints:  -. wn 2   -> 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-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-cleq 1877  df-clel 1880  df-ral 2109

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