Theorem reuuni4 1959

Description: Derive the property of 'the unique element in A such that ph ' when expressed explicitly as U.{x e. A | ph}.

Assertion

Ref	Expression
reuuni4	|- (E!x e. A ph -> [U.{x e. A | ph} / x]ph)

Distinct variable group(s):   x,A

Proof of Theorem reuuni4

Step	Hyp	Ref	Expression
1		reucl 1957	. 2 |- (E!x e. A ph -> U.{x e. A | ph} e. A)
2		reurex 1337	. . . 4 |- (E!x e. A ph -> E.x e. A ph)
3		hbreu1 1307	. . . . 5 |- (E!x e. A ph -> A.xE!x e. A ph)
4		hbrab1 1310	. . . . . . 7 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
5	4	hbuni 1925	. . . . . 6 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
6	5	hbsbc 1446	. . . . 5 |- ((U.{x e. A | ph} e. V -> [U.{x e. A | ph} / x]ph) -> A.x(U.{x e. A | ph} e. V -> [U.{x e. A | ph} / x]ph))
7		reuuni1 1955	. . . . . . . . . . 11 |- ((x e. A /\ E!x e. A ph) -> (ph <-> U.{x e. A | ph} = x))
8		sbceq1 1443	. . . . . . . . . . . 12 |- (x = U.{x e. A | ph} -> (ph <-> [U.{x e. A | ph} / x]ph))
9	8	cleqcoms 1104	. . . . . . . . . . 11 |- (U.{x e. A | ph} = x -> (ph <-> [U.{x e. A | ph} / x]ph))
10	7, 9	syl6bi 187	. . . . . . . . . 10 |- ((x e. A /\ E!x e. A ph) -> (ph -> (ph <-> [U.{x e. A | ph} / x]ph)))
11		ibib 448	. . . . . . . . . 10 |- ((ph -> [U.{x e. A | ph} / x]ph) <-> (ph -> (ph <-> [U.{x e. A | ph} / x]ph)))
12	10, 11	sylibr 175	. . . . . . . . 9 |- ((x e. A /\ E!x e. A ph) -> (ph -> [U.{x e. A | ph} / x]ph))
13	12	exp 291	. . . . . . . 8 |- (x e. A -> (E!x e. A ph -> (ph -> [U.{x e. A | ph} / x]ph)))
14	13	com12 13	. . . . . . 7 |- (E!x e. A ph -> (x e. A -> (ph -> [U.{x e. A | ph} / x]ph)))
15	14	a1i 7	. . . . . 6 |- (U.{x e. A | ph} e. V -> (E!x e. A ph -> (x e. A -> (ph -> [U.{x e. A | ph} / x]ph))))
16	15	com4l 39	. . . . 5 |- (E!x e. A ph -> (x e. A -> (ph -> (U.{x e. A | ph} e. V -> [U.{x e. A | ph} / x]ph))))
17	3, 6, 16	r19.23ad 1285	. . . 4 |- (E!x e. A ph -> (E.x e. A ph -> (U.{x e. A | ph} e. V -> [U.{x e. A | ph} / x]ph)))
18	2, 17	mpd 46	. . 3 |- (E!x e. A ph -> (U.{x e. A | ph} e. V -> [U.{x e. A | ph} / x]ph))
19		elisset 1354	. . 3 |- (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. V)
20	18, 19	syl5 22	. 2 |- (E!x e. A ph -> (U.{x e. A | ph} e. A -> [U.{x e. A | ph} / x]ph))
21	1, 20	mpd 46	1 |- (E!x e. A ph -> [U.{x e. A | ph} / x]ph)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  E.wrex 1202  E!wreu 1203  {crab 1204  Vcvv 1348  [wsbc 1440  U.cuni 1919

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-uni 1920

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