Theorem 2eu4 1070

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 1067 for a condition under which the naive definition holds and 2exeu 1066 for a one-way implication. See 2eu5 1071 for an alternate definition.

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 group(s):   x,y,z,w   ph,z,w

Proof of Theorem 2eu4

Step	Hyp	Ref	Expression
1		ax-17 925	. . . 4 |- (E.yph -> A.zE.yph)
2	1	eu3 1024	. . 3 |- (E!xE.yph <-> (E.xE.yph /\ E.zA.x(E.yph -> x = z)))
3		ax-17 925	. . . 4 |- (E.xph -> A.wE.xph)
4	3	eu3 1024	. . 3 |- (E!yE.xph <-> (E.yE.xph /\ E.wA.y(E.xph -> y = w)))
5	2, 4	anbi12i 369	. 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 388	. 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 728	. . . . 5 |- (E.yE.xph <-> E.xE.yph)
8	7	anbi2i 367	. . . 4 |- ((E.xE.yph /\ E.yE.xph) <-> (E.xE.yph /\ E.xE.yph))
9		anidm 331	. . . 4 |- ((E.xE.yph /\ E.xE.yph) <-> E.xE.yph)
10	8, 9	bitr 151	. . 3 |- ((E.xE.yph /\ E.yE.xph) <-> E.xE.yph)
11		hba1 698	. . . . . . . . . 10 |- (A.xA.y(ph -> y = w) -> A.xA.xA.y(ph -> y = w))
12	11	19.3r 714	. . . . . . . . 9 |- (A.xA.y(ph -> y = w) <-> A.xA.xA.y(ph -> y = w))
13	12	anbi2i 367	. . . . . . . 8 |- ((A.xA.y(ph -> x = z) /\ A.xA.y(ph -> y = w)) <-> (A.xA.y(ph -> x = z) /\ A.xA.xA.y(ph -> y = w)))
14		jcab 454	. . . . . . . . . . . 12 |- ((ph -> (x = z /\ y = w)) <-> ((ph -> x = z) /\ (ph -> y = w)))
15	14	bial 695	. . . . . . . . . . 11 |- (A.y(ph -> (x = z /\ y = w)) <-> A.y((ph -> x = z) /\ (ph -> y = w)))
16		19.26 749	. . . . . . . . . . 11 |- (A.y((ph -> x = z) /\ (ph -> y = w)) <-> (A.y(ph -> x = z) /\ A.y(ph -> y = w)))
17	15, 16	bitr 151	. . . . . . . . . 10 |- (A.y(ph -> (x = z /\ y = w)) <-> (A.y(ph -> x = z) /\ A.y(ph -> y = w)))
18	17	bial 695	. . . . . . . . 9 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> A.x(A.y(ph -> x = z) /\ A.y(ph -> y = w)))
19		19.26 749	. . . . . . . . 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)))
20	18, 19	bitr 151	. . . . . . . 8 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> (A.xA.y(ph -> x = z) /\ A.xA.y(ph -> y = w)))
21		19.26 749	. . . . . . . 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)))
22	13, 20, 21	3bitr4 158	. . . . . . 7 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> A.x(A.y(ph -> x = z) /\ A.xA.y(ph -> y = w)))
23		19.26 749	. . . . . . . . 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		ax-4 673	. . . . . . . . . . 11 |- (A.yA.y(ph -> x = z) -> A.y(ph -> x = z))
25		hba1 698	. . . . . . . . . . 11 |- (A.y(ph -> x = z) -> A.yA.y(ph -> x = z))
26	24, 25	impbi 139	. . . . . . . . . 10 |- (A.yA.y(ph -> x = z) <-> A.y(ph -> x = z))
27		alcom 715	. . . . . . . . . 10 |- (A.yA.x(ph -> y = w) <-> A.xA.y(ph -> y = w))
28	26, 27	anbi12i 369	. . . . . . . . 9 |- ((A.yA.y(ph -> x = z) /\ A.yA.x(ph -> y = w)) <-> (A.y(ph -> x = z) /\ A.xA.y(ph -> y = w)))
29	23, 28	bitr 151	. . . . . . . 8 |- (A.y(A.y(ph -> x = z) /\ A.x(ph -> y = w)) <-> (A.y(ph -> x = z) /\ A.xA.y(ph -> y = w)))
30	29	bial 695	. . . . . . 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)))
31	22, 30	bitr4 154	. . . . . 6 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.y(A.y(ph -> x = z) /\ A.x(ph -> y = w)))
32		19.23v 950	. . . . . . . 8 |- (A.y(ph -> x = z) <-> (E.yph -> x = z))
33		19.23v 950	. . . . . . . 8 |- (A.x(ph -> y = w) <-> (E.xph -> y = w))
34	32, 33	anbi12i 369	. . . . . . 7 |- ((A.y(ph -> x = z) /\ A.x(ph -> y = w)) <-> ((E.yph -> x = z) /\ (E.xph -> y = w)))
35	34	bi2al 696	. . . . . 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)))
36		hbe1 709	. . . . . . . 8 |- (E.yph -> A.yE.yph)
37		ax-17 925	. . . . . . . 8 |- (x = z -> A.y x = z)
38	36, 37	hbim 702	. . . . . . 7 |- ((E.yph -> x = z) -> A.y(E.yph -> x = z))
39		hbe1 709	. . . . . . . 8 |- (E.xph -> A.xE.xph)
40		ax-17 925	. . . . . . . 8 |- (y = w -> A.x y = w)
41	39, 40	hbim 702	. . . . . . 7 |- ((E.xph -> y = w) -> A.x(E.xph -> y = w))
42	38, 41	aaan 794	. . . . . 6 |- (A.xA.y((E.yph -> x = z) /\ (E.xph -> y = w)) <-> (A.x(E.yph -> x = z) /\ A.y(E.xph -> y = w)))
43	31, 35, 42	3bitr 155	. . . . 5 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> (A.x(E.yph -> x = z) /\ A.y(E.xph -> y = w)))
44	43	bi2ex 734	. . . 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)))
45		eeanv 980	. . . 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)))
46	44, 45	bitr2 152	. . 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)))
47	10, 46	anbi12i 369	. 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))))
48	5, 6, 47	3bitr 155	1 |- ((E!xE.yph /\ E!yE.xph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797  E!weu 1007

This theorem is referenced by:  2eu5 1071

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009

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