Theorem ralcom2 2244

Description: Commutation of restricted quantifiers. Note that x and y needn't be distinct (this makes the proof longer but illustrates the use of dvelim 1743). (The proof was shortened by Andrew Salmon, 30-May-2011.)

Assertion

Ref	Expression
ralcom2	|- (A.x e. A A.y e. A ph -> A.y e. A A.x e. A ph)

Distinct variable groups:   y,A   x,A

Proof of Theorem ralcom2

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . . . 7 |- (x = y -> (x e. A <-> y e. A))
2	1	a4s 1330	. . . . . 6 |- (A.x x = y -> (x e. A <-> y e. A))
3	2	imbi1d 675	. . . . . . . 8 |- (A.x x = y -> ((x e. A -> ph) <-> (y e. A -> ph)))
4	3	dral1 1515	. . . . . . 7 |- (A.x x = y -> (A.x(x e. A -> ph) <-> A.y(y e. A -> ph)))
5	4	bicomd 580	. . . . . 6 |- (A.x x = y -> (A.y(y e. A -> ph) <-> A.x(x e. A -> ph)))
6	2, 5	imbi12d 688	. . . . 5 |- (A.x x = y -> ((x e. A -> A.y(y e. A -> ph)) <-> (y e. A -> A.x(x e. A -> ph))))
7	6	dral1 1515	. . . 4 |- (A.x x = y -> (A.x(x e. A -> A.y(y e. A -> ph)) <-> A.y(y e. A -> A.x(x e. A -> ph))))
8	7	biimpd 170	. . 3 |- (A.x x = y -> (A.x(x e. A -> A.y(y e. A -> ph)) -> A.y(y e. A -> A.x(x e. A -> ph))))
9		hbnae 1507	. . . 4 |- (-. A.x x = y -> A.y -. A.x x = y)
10		hbnae 1507	. . . . 5 |- (-. A.x x = y -> A.x -. A.x x = y)
11		ax-17 1317	. . . . . . . 8 |- (z e. A -> A.y z e. A)
12		eleq1 1957	. . . . . . . 8 |- (z = x -> (z e. A <-> x e. A))
13	11, 12	dvelim 1743	. . . . . . 7 |- (-. A.y y = x -> (x e. A -> A.y x e. A))
14	13	nalequcoms 1504	. . . . . 6 |- (-. A.x x = y -> (x e. A -> A.y x e. A))
15		hba1 1350	. . . . . . 7 |- (A.y(y e. A -> ph) -> A.yA.y(y e. A -> ph))
16	15	a1i 8	. . . . . 6 |- (-. A.x x = y -> (A.y(y e. A -> ph) -> A.yA.y(y e. A -> ph)))
17	9, 14, 16	hbimd 1468	. . . . 5 |- (-. A.x x = y -> ((x e. A -> A.y(y e. A -> ph)) -> A.y(x e. A -> A.y(y e. A -> ph))))
18	10, 17	hbald 1471	. . . 4 |- (-. A.x x = y -> (A.x(x e. A -> A.y(y e. A -> ph)) -> A.yA.x(x e. A -> A.y(y e. A -> ph))))
19		ax-17 1317	. . . . . . 7 |- (z e. A -> A.x z e. A)
20		eleq1 1957	. . . . . . 7 |- (z = y -> (z e. A <-> y e. A))
21	19, 20	dvelim 1743	. . . . . 6 |- (-. A.x x = y -> (y e. A -> A.x y e. A))
22		ax-4 1319	. . . . . . . . 9 |- (A.y(y e. A -> ph) -> (y e. A -> ph))
23	22	imim2i 11	. . . . . . . 8 |- ((x e. A -> A.y(y e. A -> ph)) -> (x e. A -> (y e. A -> ph)))
24	23	com23 36	. . . . . . 7 |- ((x e. A -> A.y(y e. A -> ph)) -> (y e. A -> (x e. A -> ph)))
25	24	al2imi 1341	. . . . . 6 |- (A.x(x e. A -> A.y(y e. A -> ph)) -> (A.x y e. A -> A.x(x e. A -> ph)))
26	21, 25	syl9 71	. . . . 5 |- (-. A.x x = y -> (A.x(x e. A -> A.y(y e. A -> ph)) -> (y e. A -> A.x(x e. A -> ph))))
27	26	al2imi 1341	. . . 4 |- (A.y -. A.x x = y -> (A.yA.x(x e. A -> A.y(y e. A -> ph)) -> A.y(y e. A -> A.x(x e. A -> ph))))
28	9, 18, 27	sylsyld 32	. . 3 |- (-. A.x x = y -> (A.x(x e. A -> A.y(y e. A -> ph)) -> A.y(y e. A -> A.x(x e. A -> ph))))
29	8, 28	pm2.61i 140	. 2 |- (A.x(x e. A -> A.y(y e. A -> ph)) -> A.y(y e. A -> A.x(x e. A -> ph)))
30		df-ral 2109	. . 3 |- (A.x e. A A.y e. A ph <-> A.x(x e. A -> A.y e. A ph))
31		df-ral 2109	. . . . 5 |- (A.y e. A ph <-> A.y(y e. A -> ph))
32	31	imbi2i 202	. . . 4 |- ((x e. A -> A.y e. A ph) <-> (x e. A -> A.y(y e. A -> ph)))
33	32	albii 1346	. . 3 |- (A.x(x e. A -> A.y e. A ph) <-> A.x(x e. A -> A.y(y e. A -> ph)))
34	30, 33	bitri 190	. 2 |- (A.x e. A A.y e. A ph <-> A.x(x e. A -> A.y(y e. A -> ph)))
35		df-ral 2109	. . 3 |- (A.y e. A A.x e. A ph <-> A.y(y e. A -> A.x e. A ph))
36		df-ral 2109	. . . . 5 |- (A.x e. A ph <-> A.x(x e. A -> ph))
37	36	imbi2i 202	. . . 4 |- ((y e. A -> A.x e. A ph) <-> (y e. A -> A.x(x e. A -> ph)))
38	37	albii 1346	. . 3 |- (A.y(y e. A -> A.x e. A ph) <-> A.y(y e. A -> A.x(x e. A -> ph)))
39	35, 38	bitri 190	. 2 |- (A.y e. A A.x e. A ph <-> A.y(y e. A -> A.x(x e. A -> ph)))
40	29, 34, 39	3imtr4i 236	1 |- (A.x e. A A.y e. A ph -> A.y e. A A.x e. A ph)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105

This theorem is referenced by:  tz7.48lem 5164  tratrb 5831  tratrbVD 16685

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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