Theorem reu3 1902

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 1627	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
2		df-eu 1359	. 2 |- (E!x(x e. A /\ ph) <-> E.yA.x((x e. A /\ ph) <-> x = y))
3		19.28v 1281	. . . . 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 1510	. . . . . . . . . . . 12 |- (x = y -> (x e. A <-> y e. A))
5		sbequ12 1164	. . . . . . . . . . . 12 |- (x = y -> (ph <-> [y / x]ph))
6	4, 5	anbi12d 626	. . . . . . . . . . 11 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ [y / x]ph)))
7		eqeq1 1457	. . . . . . . . . . 11 |- (x = y -> (x = y <-> y = y))
8	6, 7	bibi12d 627	. . . . . . . . . 10 |- (x = y -> (((x e. A /\ ph) <-> x = y) <-> ((y e. A /\ [y / x]ph) <-> y = y)))
9		eqid 1452	. . . . . . . . . . . 12 |- y = y
10	9	tbt 717	. . . . . . . . . . 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	a4b 1191	. . . . . . . 8 |- (A.x((x e. A /\ ph) <-> x = y) -> y e. A)
15		bi1 148	. . . . . . . . . . . . 13 |- (((x e. A /\ ph) <-> x = y) -> ((x e. A /\ ph) -> x = y))
16	15	exp3a 375	. . . . . . . . . . . 12 |- (((x e. A /\ ph) <-> x = y) -> (x e. A -> (ph -> x = y)))
17	16	imp 350	. . . . . . . . . . 11 |- ((((x e. A /\ ph) <-> x = y) /\ x e. A) -> (ph -> x = y))
18		bi2 149	. . . . . . . . . . . . 13 |- (((x e. A /\ ph) <-> x = y) -> (x = y -> (x e. A /\ ph)))
19		pm3.27 323	. . . . . . . . . . . . 13 |- ((x e. A /\ ph) -> ph)
20	18, 19	syl6 22	. . . . . . . . . . . 12 |- (((x e. A /\ ph) <-> x = y) -> (x = y -> ph))
21	20	adantr 389	. . . . . . . . . . 11 |- ((((x e. A /\ ph) <-> x = y) /\ x e. A) -> (x = y -> ph))
22	17, 21	impbid 514	. . . . . . . . . 10 |- ((((x e. A /\ ph) <-> x = y) /\ x e. A) -> (ph <-> x = y))
23	22	ex 373	. . . . . . . . 9 |- (((x e. A /\ ph) <-> x = y) -> (x e. A -> (ph <-> x = y)))
24	23	a4s 960	. . . . . . . 8 |- (A.x((x e. A /\ ph) <-> x = y) -> (x e. A -> (ph <-> x = y)))
25	14, 24	jca 288	. . . . . . 7 |- (A.x((x e. A /\ ph) <-> x = y) -> (y e. A /\ (x e. A -> (ph <-> x = y))))
26	25	a5i 965	. . . . . 6 |- (A.x((x e. A /\ ph) <-> x = y) -> A.x(y e. A /\ (x e. A -> (ph <-> x = y))))
27		bi1 148	. . . . . . . . . . 11 |- ((ph <-> x = y) -> (ph -> x = y))
28	27	imim2i 17	. . . . . . . . . 10 |- ((x e. A -> (ph <-> x = y)) -> (x e. A -> (ph -> x = y)))
29	28	imp3a 361	. . . . . . . . 9 |- ((x e. A -> (ph <-> x = y)) -> ((x e. A /\ ph) -> x = y))
30	29	adantl 388	. . . . . . . 8 |- ((y e. A /\ (x e. A -> (ph <-> x = y))) -> ((x e. A /\ ph) -> x = y))
31		eleq1a 1519	. . . . . . . . . . . 12 |- (y e. A -> (x = y -> x e. A))
32	31	adantr 389	. . . . . . . . . . 11 |- ((y e. A /\ (x e. A -> (ph <-> x = y))) -> (x = y -> x e. A))
33	32	imp 350	. . . . . . . . . 10 |- (((y e. A /\ (x e. A -> (ph <-> x = y))) /\ x = y) -> x e. A)
34		bi2 149	. . . . . . . . . . . . . 14 |- ((ph <-> x = y) -> (x = y -> ph))
35	34	imim2i 17	. . . . . . . . . . . . 13 |- ((x e. A -> (ph <-> x = y)) -> (x e. A -> (x = y -> ph)))
36	35	com23 32	. . . . . . . . . . . 12 |- ((x e. A -> (ph <-> x = y)) -> (x = y -> (x e. A -> ph)))
37	36	imp 350	. . . . . . . . . . 11 |- (((x e. A -> (ph <-> x = y)) /\ x = y) -> (x e. A -> ph))
38	37	adantll 392	. . . . . . . . . 10 |- (((y e. A /\ (x e. A -> (ph <-> x = y))) /\ x = y) -> (x e. A -> ph))
39	33, 38	jcai 289	. . . . . . . . 9 |- (((y e. A /\ (x e. A -> (ph <-> x = y))) /\ x = y) -> (x e. A /\ ph))
40	39	ex 373	. . . . . . . 8 |- ((y e. A /\ (x e. A -> (ph <-> x = y))) -> (x = y -> (x e. A /\ ph)))
41	30, 40	impbid 514	. . . . . . 7 |- ((y e. A /\ (x e. A -> (ph <-> x = y))) -> ((x e. A /\ ph) <-> x = y))
42	41	19.20i 968	. . . . . 6 |- (A.x(y e. A /\ (x e. A -> (ph <-> x = y))) -> A.x((x e. A /\ ph) <-> x = y))
43	26, 42	impbi 157	. . . . 5 |- (A.x((x e. A /\ ph) <-> x = y) <-> A.x(y e. A /\ (x e. A -> (ph <-> x = y))))
44		df-ral 1625	. . . . . 6 |- (A.x e. A (ph <-> x = y) <-> A.x(x e. A -> (ph <-> x = y)))
45	44	anbi2i 479	. . . . 5 |- ((y e. A /\ A.x e. A (ph <-> x = y)) <-> (y e. A /\ A.x(x e. A -> (ph <-> x = y))))
46	3, 43, 45	3bitr4 183	. . . 4 |- (A.x((x e. A /\ ph) <-> x = y) <-> (y e. A /\ A.x e. A (ph <-> x = y)))
47	46	exbii 1027	. . 3 |- (E.yA.x((x e. A /\ ph) <-> x = y) <-> E.y(y e. A /\ A.x e. A (ph <-> x = y)))
48		df-rex 1626	. . 3 |- (E.y e. A A.x e. A (ph <-> x = y) <-> E.y(y e. A /\ A.x e. A (ph <-> x = y)))
49	47, 48	bitr4 176	. 2 |- (E.yA.x((x e. A /\ ph) <-> x = y) <-> E.y e. A A.x e. A (ph <-> x = y))
50	1, 2, 49	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 950  E.wex 956   = wceq 1099   e. wcel 1105  [wsbc 1153  E!weu 1357  A.wral 1621  E.wrex 1622  E!wreu 1623

This theorem is referenced by:  reu6 1903  reu8 1907

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-gen 955  ax-9 1102  ax-17 1190  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-cleq 1446  df-clel 1449  df-ral 1625  df-rex 1626  df-reu 1627

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