Theorem reuuni4 2882

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:   x,A

Proof of Theorem reuuni4

Step	Hyp	Ref	Expression
1		reucl 2880	. 2 |- (E!x e. A ph -> U.{x e. A | ph} e. A)
2		reurex 1924	. . . 4 |- (E!x e. A ph -> E.x e. A ph)
3		hbreu1 1765	. . . . 5 |- (E!x e. A ph -> A.xE!x e. A ph)
4		hbrab1 1769	. . . . . . 7 |- (y e. {x e. A | ph} -> A.x y e. {x e. A | ph})
5	4	hbuni 2504	. . . . . 6 |- (y e. U.{x e. A | ph} -> A.x y e. U.{x e. A | ph})
6	5	hbsbc1 1945	. . . . 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 2877	. . . . . . . . . 10 |- ((x e. A /\ E!x e. A ph) -> (ph <-> U.{x e. A | ph} = x))
8		sbceq1a 1940	. . . . . . . . . . 11 |- (x = U.{x e. A | ph} -> (ph <-> [U.{x e. A | ph} / x]ph))
9	8	eqcoms 1475	. . . . . . . . . 10 |- (U.{x e. A | ph} = x -> (ph <-> [U.{x e. A | ph} / x]ph))
10	7, 9	syl6bi 214	. . . . . . . . 9 |- ((x e. A /\ E!x e. A ph) -> (ph -> (ph <-> [U.{x e. A | ph} / x]ph)))
11	10	ibd 593	. . . . . . . 8 |- ((x e. A /\ E!x e. A ph) -> (ph -> [U.{x e. A | ph} / x]ph))
12	11	expcom 374	. . . . . . 7 |- (E!x e. A ph -> (x e. A -> (ph -> [U.{x e. A | ph} / x]ph)))
13	12	a1i 8	. . . . . 6 |- (U.{x e. A | ph} e. V -> (E!x e. A ph -> (x e. A -> (ph -> [U.{x e. A | ph} / x]ph))))
14	13	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))))
15	3, 6, 14	r19.23ad 1742	. . . 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)))
16	2, 15	mpd 26	. . 3 |- (E!x e. A ph -> (U.{x e. A | ph} e. V -> [U.{x e. A | ph} / x]ph))
17		elisset 1813	. . 3 |- (U.{x e. A | ph} e. A -> U.{x e. A | ph} e. V)
18	16, 17	syl5 21	. 2 |- (E!x e. A ph -> (U.{x e. A | ph} e. A -> [U.{x e. A | ph} / x]ph))
19	1, 18	mpd 26	1 |- (E!x e. A ph -> [U.{x e. A | ph} / x]ph)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  [wsbc 1168  E.wrex 1643  E!wreu 1644  {crab 1645  Vcvv 1807  U.cuni 2498

This theorem is referenced by:  reucl2 2883  reuuniss 2884  reuuniss2 2886

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-uni 2499

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