Theorem 2mo 1851

Description: Two equivalent expressions for double "at most one."

Assertion

Ref	Expression
2mo	|- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))

Distinct variable groups:   x,y,z,w   ph,z,w

Proof of Theorem 2mo

Step	Hyp	Ref	Expression
1		equequ2 1495	. . . . . . 7 |- (v = z -> (x = v <-> x = z))
2		equequ2 1495	. . . . . . 7 |- (u = w -> (y = u <-> y = w))
3	1, 2	bi2anan9 694	. . . . . 6 |- ((v = z /\ u = w) -> ((x = v /\ y = u) <-> (x = z /\ y = w)))
4	3	imbi2d 674	. . . . 5 |- ((v = z /\ u = w) -> ((ph -> (x = v /\ y = u)) <-> (ph -> (x = z /\ y = w))))
5	4	2albidv 1658	. . . 4 |- ((v = z /\ u = w) -> (A.xA.y(ph -> (x = v /\ y = u)) <-> A.xA.y(ph -> (x = z /\ y = w))))
6	5	cbvex2v 1701	. . 3 |- (E.vE.uA.xA.y(ph -> (x = v /\ y = u)) <-> E.zE.wA.xA.y(ph -> (x = z /\ y = w)))
7		ax-17 1317	. . . . . . . . 9 |- ((ph -> (x = v /\ y = u)) -> A.z(ph -> (x = v /\ y = u)))
8		ax-17 1317	. . . . . . . . 9 |- ((ph -> (x = v /\ y = u)) -> A.w(ph -> (x = v /\ y = u)))
9		hbs1 1722	. . . . . . . . . 10 |- ([z / x][w / y]ph -> A.x[z / x][w / y]ph)
10		ax-17 1317	. . . . . . . . . 10 |- ((z = v /\ w = u) -> A.x(z = v /\ w = u))
11	9, 10	hbim 1354	. . . . . . . . 9 |- (([z / x][w / y]ph -> (z = v /\ w = u)) -> A.x([z / x][w / y]ph -> (z = v /\ w = u)))
12		hbs1 1722	. . . . . . . . . . 11 |- ([w / y]ph -> A.y[w / y]ph)
13	12	hbsb 1723	. . . . . . . . . 10 |- ([z / x][w / y]ph -> A.y[z / x][w / y]ph)
14		ax-17 1317	. . . . . . . . . 10 |- ((z = v /\ w = u) -> A.y(z = v /\ w = u))
15	13, 14	hbim 1354	. . . . . . . . 9 |- (([z / x][w / y]ph -> (z = v /\ w = u)) -> A.y([z / x][w / y]ph -> (z = v /\ w = u)))
16		sbequ12 1545	. . . . . . . . . . 11 |- (y = w -> (ph <-> [w / y]ph))
17		sbequ12 1545	. . . . . . . . . . 11 |- (x = z -> ([w / y]ph <-> [z / x][w / y]ph))
18	16, 17	sylan9bbr 600	. . . . . . . . . 10 |- ((x = z /\ y = w) -> (ph <-> [z / x][w / y]ph))
19		equequ1 1494	. . . . . . . . . . 11 |- (x = z -> (x = v <-> z = v))
20		equequ1 1494	. . . . . . . . . . 11 |- (y = w -> (y = u <-> w = u))
21	19, 20	bi2anan9 694	. . . . . . . . . 10 |- ((x = z /\ y = w) -> ((x = v /\ y = u) <-> (z = v /\ w = u)))
22	18, 21	imbi12d 688	. . . . . . . . 9 |- ((x = z /\ y = w) -> ((ph -> (x = v /\ y = u)) <-> ([z / x][w / y]ph -> (z = v /\ w = u))))
23	7, 8, 11, 15, 22	cbval2 1698	. . . . . . . 8 |- (A.xA.y(ph -> (x = v /\ y = u)) <-> A.zA.w([z / x][w / y]ph -> (z = v /\ w = u)))
24	23	biimpi 168	. . . . . . 7 |- (A.xA.y(ph -> (x = v /\ y = u)) -> A.zA.w([z / x][w / y]ph -> (z = v /\ w = u)))
25	24	ancli 320	. . . . . 6 |- (A.xA.y(ph -> (x = v /\ y = u)) -> (A.xA.y(ph -> (x = v /\ y = u)) /\ A.zA.w([z / x][w / y]ph -> (z = v /\ w = u))))
26		alcom 1379	. . . . . . . . 9 |- (A.yA.zA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) <-> A.zA.yA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))))
27	8, 15	aaan 1477	. . . . . . . . . 10 |- (A.yA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) <-> (A.y(ph -> (x = v /\ y = u)) /\ A.w([z / x][w / y]ph -> (z = v /\ w = u))))
28	27	albii 1346	. . . . . . . . 9 |- (A.zA.yA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) <-> A.z(A.y(ph -> (x = v /\ y = u)) /\ A.w([z / x][w / y]ph -> (z = v /\ w = u))))
29	26, 28	bitri 190	. . . . . . . 8 |- (A.yA.zA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) <-> A.z(A.y(ph -> (x = v /\ y = u)) /\ A.w([z / x][w / y]ph -> (z = v /\ w = u))))
30	29	albii 1346	. . . . . . 7 |- (A.xA.yA.zA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) <-> A.xA.z(A.y(ph -> (x = v /\ y = u)) /\ A.w([z / x][w / y]ph -> (z = v /\ w = u))))
31		ax-17 1317	. . . . . . . 8 |- (A.y(ph -> (x = v /\ y = u)) -> A.zA.y(ph -> (x = v /\ y = u)))
32	11	hbal 1352	. . . . . . . 8 |- (A.w([z / x][w / y]ph -> (z = v /\ w = u)) -> A.xA.w([z / x][w / y]ph -> (z = v /\ w = u)))
33	31, 32	aaan 1477	. . . . . . 7 |- (A.xA.z(A.y(ph -> (x = v /\ y = u)) /\ A.w([z / x][w / y]ph -> (z = v /\ w = u))) <-> (A.xA.y(ph -> (x = v /\ y = u)) /\ A.zA.w([z / x][w / y]ph -> (z = v /\ w = u))))
34	30, 33	bitri 190	. . . . . 6 |- (A.xA.yA.zA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) <-> (A.xA.y(ph -> (x = v /\ y = u)) /\ A.zA.w([z / x][w / y]ph -> (z = v /\ w = u))))
35	25, 34	sylibr 217	. . . . 5 |- (A.xA.y(ph -> (x = v /\ y = u)) -> A.xA.yA.zA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))))
36		prth 615	. . . . . . . 8 |- (((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) -> ((ph /\ [z / x][w / y]ph) -> ((x = v /\ y = u) /\ (z = v /\ w = u))))
37		equtr2 1492	. . . . . . . . . 10 |- ((x = v /\ z = v) -> x = z)
38		equtr2 1492	. . . . . . . . . 10 |- ((y = u /\ w = u) -> y = w)
39	37, 38	anim12i 360	. . . . . . . . 9 |- (((x = v /\ z = v) /\ (y = u /\ w = u)) -> (x = z /\ y = w))
40	39	an4s 566	. . . . . . . 8 |- (((x = v /\ y = u) /\ (z = v /\ w = u)) -> (x = z /\ y = w))
41	36, 40	syl6 25	. . . . . . 7 |- (((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) -> ((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))
42	41	2alimi 1339	. . . . . 6 |- (A.zA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) -> A.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))
43	42	2alimi 1339	. . . . 5 |- (A.xA.yA.zA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) -> A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))
44	35, 43	syl 12	. . . 4 |- (A.xA.y(ph -> (x = v /\ y = u)) -> A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))
45	44	19.23aivv 1675	. . 3 |- (E.vE.uA.xA.y(ph -> (x = v /\ y = u)) -> A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))
46	6, 45	sylbir 218	. 2 |- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) -> A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))
47		alrot4 1451	. . . . . . 7 |- (A.xA.yA.zA.w([z / x][w / y]ph -> (ph -> (x = z /\ y = w))) <-> A.zA.wA.xA.y([z / x][w / y]ph -> (ph -> (x = z /\ y = w))))
48		alim 1340	. . . . . . . . 9 |- (A.y([z / x][w / y]ph -> (ph -> (x = z /\ y = w))) -> (A.y[z / x][w / y]ph -> A.y(ph -> (x = z /\ y = w))))
49	48	al2imi 1341	. . . . . . . 8 |- (A.xA.y([z / x][w / y]ph -> (ph -> (x = z /\ y = w))) -> (A.xA.y[z / x][w / y]ph -> A.xA.y(ph -> (x = z /\ y = w))))
50	49	2alimi 1339	. . . . . . 7 |- (A.zA.wA.xA.y([z / x][w / y]ph -> (ph -> (x = z /\ y = w))) -> A.zA.w(A.xA.y[z / x][w / y]ph -> A.xA.y(ph -> (x = z /\ y = w))))
51	47, 50	sylbi 216	. . . . . 6 |- (A.xA.yA.zA.w([z / x][w / y]ph -> (ph -> (x = z /\ y = w))) -> A.zA.w(A.xA.y[z / x][w / y]ph -> A.xA.y(ph -> (x = z /\ y = w))))
52		exim 1386	. . . . . . 7 |- (A.w(A.xA.y[z / x][w / y]ph -> A.xA.y(ph -> (x = z /\ y = w))) -> (E.wA.xA.y[z / x][w / y]ph -> E.wA.xA.y(ph -> (x = z /\ y = w))))
53	52	alimi 1338	. . . . . 6 |- (A.zA.w(A.xA.y[z / x][w / y]ph -> A.xA.y(ph -> (x = z /\ y = w))) -> A.z(E.wA.xA.y[z / x][w / y]ph -> E.wA.xA.y(ph -> (x = z /\ y = w))))
54		exim 1386	. . . . . 6 |- (A.z(E.wA.xA.y[z / x][w / y]ph -> E.wA.xA.y(ph -> (x = z /\ y = w))) -> (E.zE.wA.xA.y[z / x][w / y]ph -> E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
55	51, 53, 54	3syl 24	. . . . 5 |- (A.xA.yA.zA.w([z / x][w / y]ph -> (ph -> (x = z /\ y = w))) -> (E.zE.wA.xA.y[z / x][w / y]ph -> E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
56	9, 13	19.21ai 1345	. . . . . 6 |- ([z / x][w / y]ph -> A.xA.y[z / x][w / y]ph)
57	56	2eximi 1388	. . . . 5 |- (E.zE.w[z / x][w / y]ph -> E.zE.wA.xA.y[z / x][w / y]ph)
58	55, 57	syl5com 63	. . . 4 |- (E.zE.w[z / x][w / y]ph -> (A.xA.yA.zA.w([z / x][w / y]ph -> (ph -> (x = z /\ y = w))) -> E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
59		impexp 374	. . . . . . 7 |- (((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)) <-> (ph -> ([z / x][w / y]ph -> (x = z /\ y = w))))
60		bi2.04 177	. . . . . . 7 |- ((ph -> ([z / x][w / y]ph -> (x = z /\ y = w))) <-> ([z / x][w / y]ph -> (ph -> (x = z /\ y = w))))
61	59, 60	bitri 190	. . . . . 6 |- (((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)) <-> ([z / x][w / y]ph -> (ph -> (x = z /\ y = w))))
62	61	2albii 1347	. . . . 5 |- (A.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)) <-> A.zA.w([z / x][w / y]ph -> (ph -> (x = z /\ y = w))))
63	62	2albii 1347	. . . 4 |- (A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)) <-> A.xA.yA.zA.w([z / x][w / y]ph -> (ph -> (x = z /\ y = w))))
64	58, 63	syl5ib 223	. . 3 |- (E.zE.w[z / x][w / y]ph -> (A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)) -> E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
65		alnex 1380	. . . . . . 7 |- (A.w -. [z / x][w / y]ph <-> -. E.w[z / x][w / y]ph)
66	65	albii 1346	. . . . . 6 |- (A.zA.w -. [z / x][w / y]ph <-> A.z -. E.w[z / x][w / y]ph)
67		alnex 1380	. . . . . 6 |- (A.z -. E.w[z / x][w / y]ph <-> -. E.zE.w[z / x][w / y]ph)
68	66, 67	bitri 190	. . . . 5 |- (A.zA.w -. [z / x][w / y]ph <-> -. E.zE.w[z / x][w / y]ph)
69		ax-17 1317	. . . . . . . 8 |- (-. ph -> A.z -. ph)
70		ax-17 1317	. . . . . . . 8 |- (-. ph -> A.w -. ph)
71	9	hbn 1351	. . . . . . . 8 |- (-. [z / x][w / y]ph -> A.x -. [z / x][w / y]ph)
72	13	hbn 1351	. . . . . . . 8 |- (-. [z / x][w / y]ph -> A.y -. [z / x][w / y]ph)
73	18	notbid 673	. . . . . . . 8 |- ((x = z /\ y = w) -> (-. ph <-> -. [z / x][w / y]ph))
74	69, 70, 71, 72, 73	cbval2 1698	. . . . . . 7 |- (A.xA.y -. ph <-> A.zA.w -. [z / x][w / y]ph)
75	74	biimpri 169	. . . . . 6 |- (A.zA.w -. [z / x][w / y]ph -> A.xA.y -. ph)
76		pm2.21 92	. . . . . . 7 |- (-. ph -> (ph -> (x = z /\ y = w)))
77	76	2alimi 1339	. . . . . 6 |- (A.xA.y -. ph -> A.xA.y(ph -> (x = z /\ y = w)))
78		19.8a 1376	. . . . . . 7 |- (E.wA.xA.y(ph -> (x = z /\ y = w)) -> E.zE.wA.xA.y(ph -> (x = z /\ y = w)))
79	78	19.23bi 1414	. . . . . 6 |- (A.xA.y(ph -> (x = z /\ y = w)) -> E.zE.wA.xA.y(ph -> (x = z /\ y = w)))
80	75, 77, 79	3syl 24	. . . . 5 |- (A.zA.w -. [z / x][w / y]ph -> E.zE.wA.xA.y(ph -> (x = z /\ y = w)))
81	68, 80	sylbir 218	. . . 4 |- (-. E.zE.w[z / x][w / y]ph -> E.zE.wA.xA.y(ph -> (x = z /\ y = w)))
82	81	a1d 15	. . 3 |- (-. E.zE.w[z / x][w / y]ph -> (A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)) -> E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
83	64, 82	pm2.61i 140	. 2 |- (A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)) -> E.zE.wA.xA.y(ph -> (x = z /\ y = w)))
84	46, 83	impbii 174	1 |- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298  E.wex 1326  [wsbc 1534

This theorem is referenced by:  2mos 1852  2eu6 1858

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-16 1580  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536

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