Theorem mo 1787

Description: Equivalent definitions of "there exists at most one."

Hypothesis

Ref	Expression
mo.1	|- (ph -> A.yph)

Assertion

Ref	Expression
mo	|- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))

Distinct variable group:   x,y

Proof of Theorem mo

Step	Hyp	Ref	Expression
1		mo.1	. . . . . 6 |- (ph -> A.yph)
2		ax-17 1317	. . . . . 6 |- (x = z -> A.y x = z)
3	1, 2	hbim 1354	. . . . 5 |- ((ph -> x = z) -> A.y(ph -> x = z))
4	3	hbal 1352	. . . 4 |- (A.x(ph -> x = z) -> A.yA.x(ph -> x = z))
5		ax-17 1317	. . . 4 |- (A.x(ph -> x = y) -> A.zA.x(ph -> x = y))
6		equequ2 1495	. . . . . 6 |- (z = y -> (x = z <-> x = y))
7	6	imbi2d 674	. . . . 5 |- (z = y -> ((ph -> x = z) <-> (ph -> x = y)))
8	7	albidv 1656	. . . 4 |- (z = y -> (A.x(ph -> x = z) <-> A.x(ph -> x = y)))
9	4, 5, 8	cbvex 1529	. . 3 |- (E.zA.x(ph -> x = z) <-> E.yA.x(ph -> x = y))
10		hbs1 1722	. . . . . . . . 9 |- ([y / x]ph -> A.x[y / x]ph)
11		ax-17 1317	. . . . . . . . 9 |- (y = z -> A.x y = z)
12	10, 11	hbim 1354	. . . . . . . 8 |- (([y / x]ph -> y = z) -> A.x([y / x]ph -> y = z))
13		sbequ2 1543	. . . . . . . . 9 |- (x = y -> ([y / x]ph -> ph))
14		ax-8 1306	. . . . . . . . 9 |- (x = y -> (x = z -> y = z))
15	13, 14	imim12d 69	. . . . . . . 8 |- (x = y -> ((ph -> x = z) -> ([y / x]ph -> y = z)))
16	3, 12, 15	cbv3 1525	. . . . . . 7 |- (A.x(ph -> x = z) -> A.y([y / x]ph -> y = z))
17	16	ancli 320	. . . . . 6 |- (A.x(ph -> x = z) -> (A.x(ph -> x = z) /\ A.y([y / x]ph -> y = z)))
18	3, 12	aaan 1477	. . . . . 6 |- (A.xA.y((ph -> x = z) /\ ([y / x]ph -> y = z)) <-> (A.x(ph -> x = z) /\ A.y([y / x]ph -> y = z)))
19	17, 18	sylibr 217	. . . . 5 |- (A.x(ph -> x = z) -> A.xA.y((ph -> x = z) /\ ([y / x]ph -> y = z)))
20		prth 615	. . . . . . 7 |- (((ph -> x = z) /\ ([y / x]ph -> y = z)) -> ((ph /\ [y / x]ph) -> (x = z /\ y = z)))
21		equtr2 1492	. . . . . . 7 |- ((x = z /\ y = z) -> x = y)
22	20, 21	syl6 25	. . . . . 6 |- (((ph -> x = z) /\ ([y / x]ph -> y = z)) -> ((ph /\ [y / x]ph) -> x = y))
23	22	2alimi 1339	. . . . 5 |- (A.xA.y((ph -> x = z) /\ ([y / x]ph -> y = z)) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
24	19, 23	syl 12	. . . 4 |- (A.x(ph -> x = z) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
25	24	19.23aiv 1674	. . 3 |- (E.zA.x(ph -> x = z) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
26	9, 25	sylbir 218	. 2 |- (E.yA.x(ph -> x = y) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
27		alim 1340	. . . . . . . 8 |- (A.x([y / x]ph -> (ph -> x = y)) -> (A.x[y / x]ph -> A.x(ph -> x = y)))
28	27	alimi 1338	. . . . . . 7 |- (A.yA.x([y / x]ph -> (ph -> x = y)) -> A.y(A.x[y / x]ph -> A.x(ph -> x = y)))
29	28	a7s 1337	. . . . . 6 |- (A.xA.y([y / x]ph -> (ph -> x = y)) -> A.y(A.x[y / x]ph -> A.x(ph -> x = y)))
30		exim 1386	. . . . . 6 |- (A.y(A.x[y / x]ph -> A.x(ph -> x = y)) -> (E.yA.x[y / x]ph -> E.yA.x(ph -> x = y)))
31	29, 30	syl 12	. . . . 5 |- (A.xA.y([y / x]ph -> (ph -> x = y)) -> (E.yA.x[y / x]ph -> E.yA.x(ph -> x = y)))
32	1	hbsb3 1575	. . . . . 6 |- ([y / x]ph -> A.x[y / x]ph)
33	32	eximi 1387	. . . . 5 |- (E.y[y / x]ph -> E.yA.x[y / x]ph)
34	31, 33	syl5com 63	. . . 4 |- (E.y[y / x]ph -> (A.xA.y([y / x]ph -> (ph -> x = y)) -> E.yA.x(ph -> x = y)))
35		impexp 374	. . . . . 6 |- (((ph /\ [y / x]ph) -> x = y) <-> (ph -> ([y / x]ph -> x = y)))
36		bi2.04 177	. . . . . 6 |- ((ph -> ([y / x]ph -> x = y)) <-> ([y / x]ph -> (ph -> x = y)))
37	35, 36	bitri 190	. . . . 5 |- (((ph /\ [y / x]ph) -> x = y) <-> ([y / x]ph -> (ph -> x = y)))
38	37	2albii 1347	. . . 4 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) <-> A.xA.y([y / x]ph -> (ph -> x = y)))
39	34, 38	syl5ib 223	. . 3 |- (E.y[y / x]ph -> (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E.yA.x(ph -> x = y)))
40		alnex 1380	. . . . 5 |- (A.y -. [y / x]ph <-> -. E.y[y / x]ph)
41	32	hbn 1351	. . . . . . 7 |- (-. [y / x]ph -> A.x -. [y / x]ph)
42	1	hbn 1351	. . . . . . 7 |- (-. ph -> A.y -. ph)
43		sbequ1 1542	. . . . . . . . 9 |- (x = y -> (ph -> [y / x]ph))
44	43	equcoms 1489	. . . . . . . 8 |- (y = x -> (ph -> [y / x]ph))
45	44	con3d 111	. . . . . . 7 |- (y = x -> (-. [y / x]ph -> -. ph))
46	41, 42, 45	cbv3 1525	. . . . . 6 |- (A.y -. [y / x]ph -> A.x -. ph)
47		pm2.21 92	. . . . . . 7 |- (-. ph -> (ph -> x = y))
48	47	alimi 1338	. . . . . 6 |- (A.x -. ph -> A.x(ph -> x = y))
49		19.8a 1376	. . . . . 6 |- (A.x(ph -> x = y) -> E.yA.x(ph -> x = y))
50	46, 48, 49	3syl 24	. . . . 5 |- (A.y -. [y / x]ph -> E.yA.x(ph -> x = y))
51	40, 50	sylbir 218	. . . 4 |- (-. E.y[y / x]ph -> E.yA.x(ph -> x = y))
52	51	a1d 15	. . 3 |- (-. E.y[y / x]ph -> (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E.yA.x(ph -> x = y)))
53	39, 52	pm2.61i 140	. 2 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E.yA.x(ph -> x = y))
54	26, 53	impbii 174	1 |- (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y))

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:  eu2 1791  eu3 1792  mo3 1797

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-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

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

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