Theorem 2eu4 1455

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 1452 for a condition under which the naive definition holds and 2exeu 1449 for a one-way implication. See 2eu5 1456 and 2eu8 1459 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-17 973	. . . 4 |- (E.yph -> A.zE.yph)
2	1	eu3 1399	. . 3 |- (E!xE.yph <-> (E.xE.yph /\ E.zA.x(E.yph -> x = z)))
3		ax-17 973	. . . 4 |- (E.xph -> A.wE.xph)
4	3	eu3 1399	. . 3 |- (E!yE.xph <-> (E.yE.xph /\ E.wA.y(E.xph -> y = w)))
5	2, 4	anbi12i 484	. 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 508	. 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 1048	. . . . 5 |- (E.yE.xph <-> E.xE.yph)
8	7	anbi2i 482	. . . 4 |- ((E.xE.yph /\ E.yE.xph) <-> (E.xE.yph /\ E.xE.yph))
9		anidm 434	. . . 4 |- ((E.xE.yph /\ E.xE.yph) <-> E.xE.yph)
10	8, 9	bitr 173	. . 3 |- ((E.xE.yph /\ E.yE.xph) <-> E.xE.yph)
11		hba1 1005	. . . . . . . . . 10 |- (A.xA.y(ph -> y = w) -> A.xA.xA.y(ph -> y = w))
12	11	19.3 1033	. . . . . . . . 9 |- (A.xA.xA.y(ph -> y = w) <-> A.xA.y(ph -> y = w))
13	12	anbi2i 482	. . . . . . . 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 1069	. . . . . . . 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 600	. . . . . . . . . . . 12 |- ((ph -> (x = z /\ y = w)) <-> ((ph -> x = z) /\ (ph -> y = w)))
16	15	albii 1001	. . . . . . . . . . 11 |- (A.y(ph -> (x = z /\ y = w)) <-> A.y((ph -> x = z) /\ (ph -> y = w)))
17		19.26 1069	. . . . . . . . . . 11 |- (A.y((ph -> x = z) /\ (ph -> y = w)) <-> (A.y(ph -> x = z) /\ A.y(ph -> y = w)))
18	16, 17	bitr 173	. . . . . . . . . 10 |- (A.y(ph -> (x = z /\ y = w)) <-> (A.y(ph -> x = z) /\ A.y(ph -> y = w)))
19	18	albii 1001	. . . . . . . . 9 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> A.x(A.y(ph -> x = z) /\ A.y(ph -> y = w)))
20		19.26 1069	. . . . . . . . 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	bitr 173	. . . . . . . 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	3bitr4r 184	. . . . . . 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 1069	. . . . . . . . 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 1005	. . . . . . . . . . 11 |- (A.y(ph -> x = z) -> A.yA.y(ph -> x = z))
25	24	19.3 1033	. . . . . . . . . 10 |- (A.yA.y(ph -> x = z) <-> A.y(ph -> x = z))
26		alcom 1034	. . . . . . . . . 10 |- (A.yA.x(ph -> y = w) <-> A.xA.y(ph -> y = w))
27	25, 26	anbi12i 484	. . . . . . . . 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	bitr 173	. . . . . . . 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 1001	. . . . . . 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	bitr4 176	. . . . . 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 1295	. . . . . . . 8 |- (A.y(ph -> x = z) <-> (E.yph -> x = z))
32		19.23v 1295	. . . . . . . 8 |- (A.x(ph -> y = w) <-> (E.xph -> y = w))
33	31, 32	anbi12i 484	. . . . . . 7 |- ((A.y(ph -> x = z) /\ A.x(ph -> y = w)) <-> ((E.yph -> x = z) /\ (E.xph -> y = w)))
34	33	2albii 1002	. . . . . 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 1018	. . . . . . . 8 |- (E.yph -> A.yE.yph)
36		ax-17 973	. . . . . . . 8 |- (x = z -> A.y x = z)
37	35, 36	hbim 1009	. . . . . . 7 |- ((E.yph -> x = z) -> A.y(E.yph -> x = z))
38		hbe1 1018	. . . . . . . 8 |- (E.xph -> A.xE.xph)
39		ax-17 973	. . . . . . . 8 |- (y = w -> A.x y = w)
40	38, 39	hbim 1009	. . . . . . 7 |- ((E.xph -> y = w) -> A.x(E.xph -> y = w))
41	37, 40	aaan 1121	. . . . . 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	3bitr 177	. . . . 5 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> (A.x(E.yph -> x = z) /\ A.y(E.xph -> y = w)))
43	42	2exbii 1054	. . . 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 1325	. . . 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	bitr2 174	. . 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 484	. 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	3bitr 177	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 146   /\ wa 223  A.wal 956   = wceq 958  E.wex 982  E!weu 1382

This theorem is referenced by:  2eu5 1456  2eu6 1457

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384

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