Theorem reu6 3026

Description: A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.)

Assertion

Ref	Expression
reu6	|- (E!x e. A ph <-> E.y e. A A.x e. A (ph <-> x = y))

Distinct variable groups:   x,y,A   ph,y

Allowed substitution hint:   ph(x)

Proof of Theorem reu6

Step	Hyp	Ref	Expression
1		df-reu 2622	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
2		19.28v 1895	. . . . 5 |- (A.x(y e. A /\ (x e. A -> (ph <-> x = y))) <-> (y e. A /\ A.x(x e. A -> (ph <-> x = y))))
3		eleq1 2413	. . . . . . . . . . . 12 |- (x = y -> (x e. A <-> y e. A))
4		sbequ12 1919	. . . . . . . . . . . 12 |- (x = y -> (ph <-> [y / x]ph))
5	3, 4	anbi12d 691	. . . . . . . . . . 11 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ [y / x]ph)))
6		eqeq1 2359	. . . . . . . . . . 11 |- (x = y -> (x = y <-> y = y))
7	5, 6	bibi12d 312	. . . . . . . . . 10 |- (x = y -> (((x e. A /\ ph) <-> x = y) <-> ((y e. A /\ [y / x]ph) <-> y = y)))
8		eqid 2353	. . . . . . . . . . . 12 |- y = y
9	8	tbt 333	. . . . . . . . . . 11 |- ((y e. A /\ [y / x]ph) <-> ((y e. A /\ [y / x]ph) <-> y = y))
10		simpl 443	. . . . . . . . . . 11 |- ((y e. A /\ [y / x]ph) -> y e. A)
11	9, 10	sylbir 204	. . . . . . . . . 10 |- (((y e. A /\ [y / x]ph) <-> y = y) -> y e. A)
12	7, 11	syl6bi 219	. . . . . . . . 9 |- (x = y -> (((x e. A /\ ph) <-> x = y) -> y e. A))
13	12	spimv 1990	. . . . . . . 8 |- (A.x((x e. A /\ ph) <-> x = y) -> y e. A)
14		bi1 178	. . . . . . . . . . . 12 |- (((x e. A /\ ph) <-> x = y) -> ((x e. A /\ ph) -> x = y))
15	14	expdimp 426	. . . . . . . . . . 11 |- ((((x e. A /\ ph) <-> x = y) /\ x e. A) -> (ph -> x = y))
16		bi2 189	. . . . . . . . . . . . 13 |- (((x e. A /\ ph) <-> x = y) -> (x = y -> (x e. A /\ ph)))
17		simpr 447	. . . . . . . . . . . . 13 |- ((x e. A /\ ph) -> ph)
18	16, 17	syl6 29	. . . . . . . . . . . 12 |- (((x e. A /\ ph) <-> x = y) -> (x = y -> ph))
19	18	adantr 451	. . . . . . . . . . 11 |- ((((x e. A /\ ph) <-> x = y) /\ x e. A) -> (x = y -> ph))
20	15, 19	impbid 183	. . . . . . . . . 10 |- ((((x e. A /\ ph) <-> x = y) /\ x e. A) -> (ph <-> x = y))
21	20	ex 423	. . . . . . . . 9 |- (((x e. A /\ ph) <-> x = y) -> (x e. A -> (ph <-> x = y)))
22	21	sps 1754	. . . . . . . 8 |- (A.x((x e. A /\ ph) <-> x = y) -> (x e. A -> (ph <-> x = y)))
23	13, 22	jca 518	. . . . . . 7 |- (A.x((x e. A /\ ph) <-> x = y) -> (y e. A /\ (x e. A -> (ph <-> x = y))))
24	23	a5i 1789	. . . . . 6 |- (A.x((x e. A /\ ph) <-> x = y) -> A.x(y e. A /\ (x e. A -> (ph <-> x = y))))
25		bi1 178	. . . . . . . . . . 11 |- ((ph <-> x = y) -> (ph -> x = y))
26	25	imim2i 13	. . . . . . . . . 10 |- ((x e. A -> (ph <-> x = y)) -> (x e. A -> (ph -> x = y)))
27	26	imp3a 420	. . . . . . . . 9 |- ((x e. A -> (ph <-> x = y)) -> ((x e. A /\ ph) -> x = y))
28	27	adantl 452	. . . . . . . 8 |- ((y e. A /\ (x e. A -> (ph <-> x = y))) -> ((x e. A /\ ph) -> x = y))
29		eleq1a 2422	. . . . . . . . . . . 12 |- (y e. A -> (x = y -> x e. A))
30	29	adantr 451	. . . . . . . . . . 11 |- ((y e. A /\ (x e. A -> (ph <-> x = y))) -> (x = y -> x e. A))
31	30	imp 418	. . . . . . . . . 10 |- (((y e. A /\ (x e. A -> (ph <-> x = y))) /\ x = y) -> x e. A)
32		bi2 189	. . . . . . . . . . . . . 14 |- ((ph <-> x = y) -> (x = y -> ph))
33	32	imim2i 13	. . . . . . . . . . . . 13 |- ((x e. A -> (ph <-> x = y)) -> (x e. A -> (x = y -> ph)))
34	33	com23 72	. . . . . . . . . . . 12 |- ((x e. A -> (ph <-> x = y)) -> (x = y -> (x e. A -> ph)))
35	34	imp 418	. . . . . . . . . . 11 |- (((x e. A -> (ph <-> x = y)) /\ x = y) -> (x e. A -> ph))
36	35	adantll 694	. . . . . . . . . 10 |- (((y e. A /\ (x e. A -> (ph <-> x = y))) /\ x = y) -> (x e. A -> ph))
37	31, 36	jcai 522	. . . . . . . . 9 |- (((y e. A /\ (x e. A -> (ph <-> x = y))) /\ x = y) -> (x e. A /\ ph))
38	37	ex 423	. . . . . . . 8 |- ((y e. A /\ (x e. A -> (ph <-> x = y))) -> (x = y -> (x e. A /\ ph)))
39	28, 38	impbid 183	. . . . . . 7 |- ((y e. A /\ (x e. A -> (ph <-> x = y))) -> ((x e. A /\ ph) <-> x = y))
40	39	alimi 1559	. . . . . 6 |- (A.x(y e. A /\ (x e. A -> (ph <-> x = y))) -> A.x((x e. A /\ ph) <-> x = y))
41	24, 40	impbii 180	. . . . 5 |- (A.x((x e. A /\ ph) <-> x = y) <-> A.x(y e. A /\ (x e. A -> (ph <-> x = y))))
42		df-ral 2620	. . . . . 6 |- (A.x e. A (ph <-> x = y) <-> A.x(x e. A -> (ph <-> x = y)))
43	42	anbi2i 675	. . . . 5 |- ((y e. A /\ A.x e. A (ph <-> x = y)) <-> (y e. A /\ A.x(x e. A -> (ph <-> x = y))))
44	2, 41, 43	3bitr4i 268	. . . 4 |- (A.x((x e. A /\ ph) <-> x = y) <-> (y e. A /\ A.x e. A (ph <-> x = y)))
45	44	exbii 1582	. . 3 |- (E.yA.x((x e. A /\ ph) <-> x = y) <-> E.y(y e. A /\ A.x e. A (ph <-> x = y)))
46		df-eu 2208	. . 3 |- (E!x(x e. A /\ ph) <-> E.yA.x((x e. A /\ ph) <-> x = y))
47		df-rex 2621	. . 3 |- (E.y e. A A.x e. A (ph <-> x = y) <-> E.y(y e. A /\ A.x e. A (ph <-> x = y)))
48	45, 46, 47	3bitr4i 268	. 2 |- (E!x(x e. A /\ ph) <-> E.y e. A A.x e. A (ph <-> x = y))
49	1, 48	bitri 240	1 |- (E!x e. A ph <-> E.y e. A A.x e. A (ph <-> x = y))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642  [wsb 1648   e. wcel 1710  E!weu 2204  A.wral 2615  E.wrex 2616  E!wreu 2617

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  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-cleq 2346  df-clel 2349  df-ral 2620  df-rex 2621  df-reu 2622

This theorem is referenced by:  reu3  3027  reu6i  3028  reu8  3033