Theorem reu3 1938

Description: A way to express restricted uniqueness.

Assertion

Ref	Expression
reu3	|- (E!x e. A ph <-> E.y e. A A.x e. A (ph <-> x = y))

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

Proof of Theorem reu3

Step	Hyp	Ref	Expression
1		df-reu 1658	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
2		df-eu 1388	. 2 |- (E!x(x e. A /\ ph) <-> E.yA.x((x e. A /\ ph) <-> x = y))
3		19.28v 1305	. . . . 5 |- (A.x(y e. A /\ (x e. A -> (ph <-> x = y))) <-> (y e. A /\ A.x(x e. A -> (ph <-> x = y))))
4		eleq1 1541	. . . . . . . . . . . 12 |- (x = y -> (x e. A <-> y e. A))
5		sbequ12 1187	. . . . . . . . . . . 12 |- (x = y -> (ph <-> [y / x]ph))
6	4, 5	anbi12d 631	. . . . . . . . . . 11 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ [y / x]ph)))
7		eqeq1 1488	. . . . . . . . . . 11 |- (x = y -> (x = y <-> y = y))
8	6, 7	bibi12d 632	. . . . . . . . . 10 |- (x = y -> (((x e. A /\ ph) <-> x = y) <-> ((y e. A /\ [y / x]ph) <-> y = y)))
9		eqid 1482	. . . . . . . . . . . 12 |- y = y
10	9	tbt 724	. . . . . . . . . . 11 |- ((y e. A /\ [y / x]ph) <-> ((y e. A /\ [y / x]ph) <-> y = y))
11		pm3.26 319	. . . . . . . . . . 11 |- ((y e. A /\ [y / x]ph) -> y e. A)
12	10, 11	sylbir 201	. . . . . . . . . 10 |- (((y e. A /\ [y / x]ph) <-> y = y) -> y e. A)
13	8, 12	syl6bi 214	. . . . . . . . 9 |- (x = y -> (((x e. A /\ ph) <-> x = y) -> y e. A))
14	13	a4imv 1213	. . . . . . . 8 |- (A.x((x e. A /\ ph) <-> x = y) -> y e. A)
15		bi1 148	. . . . . . . . . . . 12 |- (((x e. A /\ ph) <-> x = y) -> ((x e. A /\ ph) -> x = y))
16	15	expdimp 377	. . . . . . . . . . 11 |- ((((x e. A /\ ph) <-> x = y) /\ x e. A) -> (ph -> x = y))
17		bi2 149	. . . . . . . . . . . . 13 |- (((x e. A /\ ph) <-> x = y) -> (x = y -> (x e. A /\ ph)))
18		pm3.27 323	. . . . . . . . . . . . 13 |- ((x e. A /\ ph) -> ph)
19	17, 18	syl6 22	. . . . . . . . . . . 12 |- (((x e. A /\ ph) <-> x = y) -> (x = y -> ph))
20	19	adantr 391	. . . . . . . . . . 11 |- ((((x e. A /\ ph) <-> x = y) /\ x e. A) -> (x = y -> ph))
21	16, 20	impbid 519	. . . . . . . . . 10 |- ((((x e. A /\ ph) <-> x = y) /\ x e. A) -> (ph <-> x = y))
22	21	ex 373	. . . . . . . . 9 |- (((x e. A /\ ph) <-> x = y) -> (x e. A -> (ph <-> x = y)))
23	22	a4s 988	. . . . . . . 8 |- (A.x((x e. A /\ ph) <-> x = y) -> (x e. A -> (ph <-> x = y)))
24	14, 23	jca 288	. . . . . . 7 |- (A.x((x e. A /\ ph) <-> x = y) -> (y e. A /\ (x e. A -> (ph <-> x = y))))
25	24	a5i 995	. . . . . 6 |- (A.x((x e. A /\ ph) <-> x = y) -> A.x(y e. A /\ (x e. A -> (ph <-> x = y))))
26		bi1 148	. . . . . . . . . . 11 |- ((ph <-> x = y) -> (ph -> x = y))
27	26	imim2i 17	. . . . . . . . . 10 |- ((x e. A -> (ph <-> x = y)) -> (x e. A -> (ph -> x = y)))
28	27	imp3a 361	. . . . . . . . 9 |- ((x e. A -> (ph <-> x = y)) -> ((x e. A /\ ph) -> x = y))
29	28	adantl 390	. . . . . . . 8 |- ((y e. A /\ (x e. A -> (ph <-> x = y))) -> ((x e. A /\ ph) -> x = y))
30		eleq1a 1550	. . . . . . . . . . . 12 |- (y e. A -> (x = y -> x e. A))
31	30	adantr 391	. . . . . . . . . . 11 |- ((y e. A /\ (x e. A -> (ph <-> x = y))) -> (x = y -> x e. A))
32	31	imp 350	. . . . . . . . . 10 |- (((y e. A /\ (x e. A -> (ph <-> x = y))) /\ x = y) -> x e. A)
33		bi2 149	. . . . . . . . . . . . . 14 |- ((ph <-> x = y) -> (x = y -> ph))
34	33	imim2i 17	. . . . . . . . . . . . 13 |- ((x e. A -> (ph <-> x = y)) -> (x e. A -> (x = y -> ph)))
35	34	com23 32	. . . . . . . . . . . 12 |- ((x e. A -> (ph <-> x = y)) -> (x = y -> (x e. A -> ph)))
36	35	imp 350	. . . . . . . . . . 11 |- (((x e. A -> (ph <-> x = y)) /\ x = y) -> (x e. A -> ph))
37	36	adantll 394	. . . . . . . . . 10 |- (((y e. A /\ (x e. A -> (ph <-> x = y))) /\ x = y) -> (x e. A -> ph))
38	32, 37	jcai 289	. . . . . . . . 9 |- (((y e. A /\ (x e. A -> (ph <-> x = y))) /\ x = y) -> (x e. A /\ ph))
39	38	ex 373	. . . . . . . 8 |- ((y e. A /\ (x e. A -> (ph <-> x = y))) -> (x = y -> (x e. A /\ ph)))
40	29, 39	impbid 519	. . . . . . 7 |- ((y e. A /\ (x e. A -> (ph <-> x = y))) -> ((x e. A /\ ph) <-> x = y))
41	40	19.20i 998	. . . . . 6 |- (A.x(y e. A /\ (x e. A -> (ph <-> x = y))) -> A.x((x e. A /\ ph) <-> x = y))
42	25, 41	impbi 157	. . . . 5 |- (A.x((x e. A /\ ph) <-> x = y) <-> A.x(y e. A /\ (x e. A -> (ph <-> x = y))))
43		df-ral 1656	. . . . . 6 |- (A.x e. A (ph <-> x = y) <-> A.x(x e. A -> (ph <-> x = y)))
44	43	anbi2i 483	. . . . 5 |- ((y e. A /\ A.x e. A (ph <-> x = y)) <-> (y e. A /\ A.x(x e. A -> (ph <-> x = y))))
45	3, 42, 44	3bitr4 183	. . . 4 |- (A.x((x e. A /\ ph) <-> x = y) <-> (y e. A /\ A.x e. A (ph <-> x = y)))
46	45	exbii 1057	. . 3 |- (E.yA.x((x e. A /\ ph) <-> x = y) <-> E.y(y e. A /\ A.x e. A (ph <-> x = y)))
47		df-rex 1657	. . 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	46, 47	bitr4 176	. 2 |- (E.yA.x((x e. A /\ ph) <-> x = y) <-> E.y e. A A.x e. A (ph <-> x = y))
49	1, 2, 48	3bitr 177	1 |- (E!x e. A ph <-> E.y e. A A.x e. A (ph <-> x = y))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 958   = wceq 960   e. wcel 962  E.wex 984  [wsbc 1176  E!weu 1386  A.wral 1652  E.wrex 1653  E!wreu 1654

This theorem is referenced by:  reu6 1939  reu8 1943

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 967  ax-17 975  ax-4 977  ax-5o 979  ax-9o 1129  ax-ext 1466

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1178  df-eu 1388  df-cleq 1476  df-clel 1479  df-ral 1656  df-rex 1657  df-reu 1658

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