Theorem ralcom2 2776

Description: Commutation of restricted quantifiers. Note that x and y needn't be distinct (this makes the proof longer). (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)

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

Allowed substitution hints:   ph(x,y)

Proof of Theorem ralcom2

Step	Hyp	Ref	Expression
1		eleq1 2413	. . . . . . 7 |- (x = y -> (x e. A <-> y e. A))
2	1	sps 1754	. . . . . 6 |- (A.x x = y -> (x e. A <-> y e. A))
3	2	imbi1d 308	. . . . . . . . 9 |- (A.x x = y -> ((x e. A -> ph) <-> (y e. A -> ph)))
4	3	dral1 1965	. . . . . . . 8 |- (A.x x = y -> (A.x(x e. A -> ph) <-> A.y(y e. A -> ph)))
5	4	bicomd 192	. . . . . . 7 |- (A.x x = y -> (A.y(y e. A -> ph) <-> A.x(x e. A -> ph)))
6		df-ral 2620	. . . . . . 7 |- (A.y e. A ph <-> A.y(y e. A -> ph))
7		df-ral 2620	. . . . . . 7 |- (A.x e. A ph <-> A.x(x e. A -> ph))
8	5, 6, 7	3bitr4g 279	. . . . . 6 |- (A.x x = y -> (A.y e. A ph <-> A.x e. A ph))
9	2, 8	imbi12d 311	. . . . 5 |- (A.x x = y -> ((x e. A -> A.y e. A ph) <-> (y e. A -> A.x e. A ph)))
10	9	dral1 1965	. . . 4 |- (A.x x = y -> (A.x(x e. A -> A.y e. A ph) <-> A.y(y e. A -> A.x e. A ph)))
11		df-ral 2620	. . . 4 |- (A.x e. A A.y e. A ph <-> A.x(x e. A -> A.y e. A ph))
12		df-ral 2620	. . . 4 |- (A.y e. A A.x e. A ph <-> A.y(y e. A -> A.x e. A ph))
13	10, 11, 12	3bitr4g 279	. . 3 |- (A.x x = y -> (A.x e. A A.y e. A ph <-> A.y e. A A.x e. A ph))
14	13	biimpd 198	. 2 |- (A.x x = y -> (A.x e. A A.y e. A ph -> A.y e. A A.x e. A ph))
15		nfnae 1956	. . . . 5 |- F/y -. A.x x = y
16		nfra2 2669	. . . . 5 |- F/yA.x e. A A.y e. A ph
17	15, 16	nfan 1824	. . . 4 |- F/y(-. A.x x = y /\ A.x e. A A.y e. A ph)
18		nfnae 1956	. . . . . . . 8 |- F/x -. A.x x = y
19		nfra1 2665	. . . . . . . 8 |- F/xA.x e. A A.y e. A ph
20	18, 19	nfan 1824	. . . . . . 7 |- F/x(-. A.x x = y /\ A.x e. A A.y e. A ph)
21		nfcvf 2512	. . . . . . . . 9 |- (-. A.x x = y -> F/_xy)
22	21	adantr 451	. . . . . . . 8 |- ((-. A.x x = y /\ A.x e. A A.y e. A ph) -> F/_xy)
23		nfcvd 2491	. . . . . . . 8 |- ((-. A.x x = y /\ A.x e. A A.y e. A ph) -> F/_xA)
24	22, 23	nfeld 2505	. . . . . . 7 |- ((-. A.x x = y /\ A.x e. A A.y e. A ph) -> F/x y e. A)
25	20, 24	nfan1 1881	. . . . . 6 |- F/x((-. A.x x = y /\ A.x e. A A.y e. A ph) /\ y e. A)
26		rsp2 2677	. . . . . . . . 9 |- (A.x e. A A.y e. A ph -> ((x e. A /\ y e. A) -> ph))
27	26	ancomsd 440	. . . . . . . 8 |- (A.x e. A A.y e. A ph -> ((y e. A /\ x e. A) -> ph))
28	27	expdimp 426	. . . . . . 7 |- ((A.x e. A A.y e. A ph /\ y e. A) -> (x e. A -> ph))
29	28	adantll 694	. . . . . 6 |- (((-. A.x x = y /\ A.x e. A A.y e. A ph) /\ y e. A) -> (x e. A -> ph))
30	25, 29	ralrimi 2696	. . . . 5 |- (((-. A.x x = y /\ A.x e. A A.y e. A ph) /\ y e. A) -> A.x e. A ph)
31	30	ex 423	. . . 4 |- ((-. A.x x = y /\ A.x e. A A.y e. A ph) -> (y e. A -> A.x e. A ph))
32	17, 31	ralrimi 2696	. . 3 |- ((-. A.x x = y /\ A.x e. A A.y e. A ph) -> A.y e. A A.x e. A ph)
33	32	ex 423	. 2 |- (-. A.x x = y -> (A.x e. A A.y e. A ph -> A.y e. A A.x e. A ph))
34	14, 33	pm2.61i 156	1 |- (A.x e. A A.y e. A ph -> A.y e. A A.x e. A ph)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710   F/_wnfc 2477  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is referenced by: (None)