Theorem 2eu4 1492

Description: This theorem provides us with a definition of double existential uniqueness ("exactly one x and exactly one y"). Naively one might think (incorrectly) that it could be defined by E!xE!yph. See 2eu1 1489 for a condition under which the naive definition holds and 2exeu 1486 for a one-way implication. See 2eu5 1493 and 2eu8 1496 for alternate definitions.

Assertion

Ref	Expression
2eu4	|- ((E!xE.yph /\ E!yE.xph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))

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

Proof of Theorem 2eu4

Step	Hyp	Ref	Expression
1		ax-17 1007	. . . 4 |- (E.yph -> A.zE.yph)
2	1	eu3 1436	. . 3 |- (E!xE.yph <-> (E.xE.yph /\ E.zA.x(E.yph -> x = z)))
3		ax-17 1007	. . . 4 |- (E.xph -> A.wE.xph)
4	3	eu3 1436	. . 3 |- (E!yE.xph <-> (E.yE.xph /\ E.wA.y(E.xph -> y = w)))
5	2, 4	anbi12i 485	. 2 |- ((E!xE.yph /\ E!yE.xph) <-> ((E.xE.yph /\ E.zA.x(E.yph -> x = z)) /\ (E.yE.xph /\ E.wA.y(E.xph -> y = w))))
6		an4 509	. 2 |- (((E.xE.yph /\ E.zA.x(E.yph -> x = z)) /\ (E.yE.xph /\ E.wA.y(E.xph -> y = w))) <-> ((E.xE.yph /\ E.yE.xph) /\ (E.zA.x(E.yph -> x = z) /\ E.wA.y(E.xph -> y = w))))
7		excom 1082	. . . . 5 |- (E.yE.xph <-> E.xE.yph)
8	7	anbi2i 483	. . . 4 |- ((E.xE.yph /\ E.yE.xph) <-> (E.xE.yph /\ E.xE.yph))
9		anidm 433	. . . 4 |- ((E.xE.yph /\ E.xE.yph) <-> E.xE.yph)
10	8, 9	bitri 171	. . 3 |- ((E.xE.yph /\ E.yE.xph) <-> E.xE.yph)
11		hba1 1039	. . . . . . . . . 10 |- (A.xA.y(ph -> y = w) -> A.xA.xA.y(ph -> y = w))
12	11	19.3 1067	. . . . . . . . 9 |- (A.xA.xA.y(ph -> y = w) <-> A.xA.y(ph -> y = w))
13	12	anbi2i 483	. . . . . . . 8 |- ((A.xA.y(ph -> x = z) /\ A.xA.xA.y(ph -> y = w)) <-> (A.xA.y(ph -> x = z) /\ A.xA.y(ph -> y = w)))
14		19.26 1103	. . . . . . . 8 |- (A.x(A.y(ph -> x = z) /\ A.xA.y(ph -> y = w)) <-> (A.xA.y(ph -> x = z) /\ A.xA.xA.y(ph -> y = w)))
15		jcab 601	. . . . . . . . . . . 12 |- ((ph -> (x = z /\ y = w)) <-> ((ph -> x = z) /\ (ph -> y = w)))
16	15	albii 1035	. . . . . . . . . . 11 |- (A.y(ph -> (x = z /\ y = w)) <-> A.y((ph -> x = z) /\ (ph -> y = w)))
17		19.26 1103	. . . . . . . . . . 11 |- (A.y((ph -> x = z) /\ (ph -> y = w)) <-> (A.y(ph -> x = z) /\ A.y(ph -> y = w)))
18	16, 17	bitri 171	. . . . . . . . . 10 |- (A.y(ph -> (x = z /\ y = w)) <-> (A.y(ph -> x = z) /\ A.y(ph -> y = w)))
19	18	albii 1035	. . . . . . . . 9 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> A.x(A.y(ph -> x = z) /\ A.y(ph -> y = w)))
20		19.26 1103	. . . . . . . . 9 |- (A.x(A.y(ph -> x = z) /\ A.y(ph -> y = w)) <-> (A.xA.y(ph -> x = z) /\ A.xA.y(ph -> y = w)))
21	19, 20	bitri 171	. . . . . . . 8 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> (A.xA.y(ph -> x = z) /\ A.xA.y(ph -> y = w)))
22	13, 14, 21	3bitr4ri 182	. . . . . . 7 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> A.x(A.y(ph -> x = z) /\ A.xA.y(ph -> y = w)))
23		19.26 1103	. . . . . . . . 9 |- (A.y(A.y(ph -> x = z) /\ A.x(ph -> y = w)) <-> (A.yA.y(ph -> x = z) /\ A.yA.x(ph -> y = w)))
24		hba1 1039	. . . . . . . . . . 11 |- (A.y(ph -> x = z) -> A.yA.y(ph -> x = z))
25	24	19.3 1067	. . . . . . . . . 10 |- (A.yA.y(ph -> x = z) <-> A.y(ph -> x = z))
26		alcom 1068	. . . . . . . . . 10 |- (A.yA.x(ph -> y = w) <-> A.xA.y(ph -> y = w))
27	25, 26	anbi12i 485	. . . . . . . . 9 |- ((A.yA.y(ph -> x = z) /\ A.yA.x(ph -> y = w)) <-> (A.y(ph -> x = z) /\ A.xA.y(ph -> y = w)))
28	23, 27	bitri 171	. . . . . . . 8 |- (A.y(A.y(ph -> x = z) /\ A.x(ph -> y = w)) <-> (A.y(ph -> x = z) /\ A.xA.y(ph -> y = w)))
29	28	albii 1035	. . . . . . 7 |- (A.xA.y(A.y(ph -> x = z) /\ A.x(ph -> y = w)) <-> A.x(A.y(ph -> x = z) /\ A.xA.y(ph -> y = w)))
30	22, 29	bitr4i 174	. . . . . 6 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.y(A.y(ph -> x = z) /\ A.x(ph -> y = w)))
31		19.23v 1331	. . . . . . . 8 |- (A.y(ph -> x = z) <-> (E.yph -> x = z))
32		19.23v 1331	. . . . . . . 8 |- (A.x(ph -> y = w) <-> (E.xph -> y = w))
33	31, 32	anbi12i 485	. . . . . . 7 |- ((A.y(ph -> x = z) /\ A.x(ph -> y = w)) <-> ((E.yph -> x = z) /\ (E.xph -> y = w)))
34	33	2albii 1036	. . . . . 6 |- (A.xA.y(A.y(ph -> x = z) /\ A.x(ph -> y = w)) <-> A.xA.y((E.yph -> x = z) /\ (E.xph -> y = w)))
35		hbe1 1052	. . . . . . . 8 |- (E.yph -> A.yE.yph)
36		ax-17 1007	. . . . . . . 8 |- (x = z -> A.y x = z)
37	35, 36	hbim 1043	. . . . . . 7 |- ((E.yph -> x = z) -> A.y(E.yph -> x = z))
38		hbe1 1052	. . . . . . . 8 |- (E.xph -> A.xE.xph)
39		ax-17 1007	. . . . . . . 8 |- (y = w -> A.x y = w)
40	38, 39	hbim 1043	. . . . . . 7 |- ((E.xph -> y = w) -> A.x(E.xph -> y = w))
41	37, 40	aaan 1155	. . . . . 6 |- (A.xA.y((E.yph -> x = z) /\ (E.xph -> y = w)) <-> (A.x(E.yph -> x = z) /\ A.y(E.xph -> y = w)))
42	30, 34, 41	3bitri 175	. . . . 5 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> (A.x(E.yph -> x = z) /\ A.y(E.xph -> y = w)))
43	42	2exbii 1088	. . . 4 |- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> E.zE.w(A.x(E.yph -> x = z) /\ A.y(E.xph -> y = w)))
44		eeanv 1361	. . . 4 |- (E.zE.w(A.x(E.yph -> x = z) /\ A.y(E.xph -> y = w)) <-> (E.zA.x(E.yph -> x = z) /\ E.wA.y(E.xph -> y = w)))
45	43, 44	bitr2i 172	. . 3 |- ((E.zA.x(E.yph -> x = z) /\ E.wA.y(E.xph -> y = w)) <-> E.zE.wA.xA.y(ph -> (x = z /\ y = w)))
46	10, 45	anbi12i 485	. 2 |- (((E.xE.yph /\ E.yE.xph) /\ (E.zA.x(E.yph -> x = z) /\ E.wA.y(E.xph -> y = w))) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
47	5, 6, 46	3bitri 175	1 |- ((E!xE.yph /\ E!yE.xph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990   = wceq 992  E.wex 1016  E!weu 1419

This theorem is referenced by:  2eu5 1493  2eu6 1494

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421

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