Theorem 2eu4 2287

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 2284 for a condition under which the naive definition holds and 2exeu 2281 for a one-way implication. See 2eu5 2288 and 2eu8 2291 for alternate definitions. (Contributed by NM, 3-Dec-2001.)

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

Allowed substitution hints:   ph(x,y)

Proof of Theorem 2eu4

Step	Hyp	Ref	Expression
1		nfv 1619	. . . 4 |- F/zE.yph
2	1	eu3 2230	. . 3 |- (E!xE.yph <-> (E.xE.yph /\ E.zA.x(E.yph -> x = z)))
3		nfv 1619	. . . 4 |- F/wE.xph
4	3	eu3 2230	. . 3 |- (E!yE.xph <-> (E.yE.xph /\ E.wA.y(E.xph -> y = w)))
5	2, 4	anbi12i 678	. 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 797	. 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 1741	. . . . 5 |- (E.yE.xph <-> E.xE.yph)
8	7	anbi2i 675	. . . 4 |- ((E.xE.yph /\ E.yE.xph) <-> (E.xE.yph /\ E.xE.yph))
9		anidm 625	. . . 4 |- ((E.xE.yph /\ E.xE.yph) <-> E.xE.yph)
10	8, 9	bitri 240	. . 3 |- ((E.xE.yph /\ E.yE.xph) <-> E.xE.yph)
11		19.26 1593	. . . . . . . 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)))
12		nfa1 1788	. . . . . . . . . . 11 |- F/xA.xA.y(ph -> y = w)
13	12	19.3 1785	. . . . . . . . . 10 |- (A.xA.xA.y(ph -> y = w) <-> A.xA.y(ph -> y = w))
14	13	anbi2i 675	. . . . . . . . 9 |- ((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)))
15		jcab 833	. . . . . . . . . . . . 13 |- ((ph -> (x = z /\ y = w)) <-> ((ph -> x = z) /\ (ph -> y = w)))
16	15	albii 1566	. . . . . . . . . . . 12 |- (A.y(ph -> (x = z /\ y = w)) <-> A.y((ph -> x = z) /\ (ph -> y = w)))
17		19.26 1593	. . . . . . . . . . . 12 |- (A.y((ph -> x = z) /\ (ph -> y = w)) <-> (A.y(ph -> x = z) /\ A.y(ph -> y = w)))
18	16, 17	bitri 240	. . . . . . . . . . 11 |- (A.y(ph -> (x = z /\ y = w)) <-> (A.y(ph -> x = z) /\ A.y(ph -> y = w)))
19	18	albii 1566	. . . . . . . . . 10 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> A.x(A.y(ph -> x = z) /\ A.y(ph -> y = w)))
20		19.26 1593	. . . . . . . . . 10 |- (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 240	. . . . . . . . 9 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> (A.xA.y(ph -> x = z) /\ A.xA.y(ph -> y = w)))
22	14, 21	bitr4i 243	. . . . . . . 8 |- ((A.xA.y(ph -> x = z) /\ A.xA.xA.y(ph -> y = w)) <-> A.xA.y(ph -> (x = z /\ y = w)))
23	11, 22	bitr2i 241	. . . . . . 7 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> A.x(A.y(ph -> x = z) /\ A.xA.y(ph -> y = w)))
24		19.26 1593	. . . . . . . . 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)))
25		nfa1 1788	. . . . . . . . . . 11 |- F/yA.y(ph -> x = z)
26	25	19.3 1785	. . . . . . . . . 10 |- (A.yA.y(ph -> x = z) <-> A.y(ph -> x = z))
27		alcom 1737	. . . . . . . . . 10 |- (A.yA.x(ph -> y = w) <-> A.xA.y(ph -> y = w))
28	26, 27	anbi12i 678	. . . . . . . . 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	24, 28	bitri 240	. . . . . . . 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	albii 1566	. . . . . . 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	23, 30	bitr4i 243	. . . . . 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 1891	. . . . . . . 8 |- (A.y(ph -> x = z) <-> (E.yph -> x = z))
33		19.23v 1891	. . . . . . . 8 |- (A.x(ph -> y = w) <-> (E.xph -> y = w))
34	32, 33	anbi12i 678	. . . . . . 7 |- ((A.y(ph -> x = z) /\ A.x(ph -> y = w)) <-> ((E.yph -> x = z) /\ (E.xph -> y = w)))
35	34	2albii 1567	. . . . . 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		nfe1 1732	. . . . . . . 8 |- F/yE.yph
37		nfv 1619	. . . . . . . 8 |- F/y x = z
38	36, 37	nfim 1813	. . . . . . 7 |- F/y(E.yph -> x = z)
39		nfe1 1732	. . . . . . . 8 |- F/xE.xph
40		nfv 1619	. . . . . . . 8 |- F/x y = w
41	39, 40	nfim 1813	. . . . . . 7 |- F/x(E.xph -> y = w)
42	38, 41	aaan 1884	. . . . . 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	3bitri 262	. . . . 5 |- (A.xA.y(ph -> (x = z /\ y = w)) <-> (A.x(E.yph -> x = z) /\ A.y(E.xph -> y = w)))
44	43	2exbii 1583	. . . 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 1913	. . . 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	bitr2i 241	. . 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 678	. 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	3bitri 262	1 |- ((E!xE.yph /\ E!yE.xph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642  E!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208

This theorem is referenced by:  2eu5  2288  2eu6  2289