Theorem reuuni1 2872

Description: A way to express "the unique element such that" (restricted quantifier version).

Assertion

Ref	Expression
reuuni1	|- ((x e. A /\ E!x e. A ph) -> (ph <-> U.{x e. A | ph} = x))

Proof of Theorem reuuni1

Step	Hyp	Ref	Expression
1		euuni 2871	. . . . . . 7 |- (E!x(x e. A /\ ph) -> ((x e. A /\ ph) <-> U.{x | (x e. A /\ ph)} = x))
2	1	biimpd 153	. . . . . 6 |- (E!x(x e. A /\ ph) -> ((x e. A /\ ph) -> U.{x | (x e. A /\ ph)} = x))
3	2	exp3a 375	. . . . 5 |- (E!x(x e. A /\ ph) -> (x e. A -> (ph -> U.{x | (x e. A /\ ph)} = x)))
4	3	impcom 351	. . . 4 |- ((x e. A /\ E!x(x e. A /\ ph)) -> (ph -> U.{x | (x e. A /\ ph)} = x))
5		pm3.27 323	. . . . . 6 |- ((x e. A /\ ph) -> ph)
6	1, 5	syl6bir 215	. . . . 5 |- (E!x(x e. A /\ ph) -> (U.{x | (x e. A /\ ph)} = x -> ph))
7	6	adantl 388	. . . 4 |- ((x e. A /\ E!x(x e. A /\ ph)) -> (U.{x | (x e. A /\ ph)} = x -> ph))
8	4, 7	impbid 514	. . 3 |- ((x e. A /\ E!x(x e. A /\ ph)) -> (ph <-> U.{x | (x e. A /\ ph)} = x))
9		df-rab 1644	. . . . 5 |- {x e. A | ph} = {x | (x e. A /\ ph)}
10	9	unieqi 2501	. . . 4 |- U.{x e. A | ph} = U.{x | (x e. A /\ ph)}
11	10	eqeq1i 1474	. . 3 |- (U.{x e. A | ph} = x <-> U.{x | (x e. A /\ ph)} = x)
12	8, 11	syl6bbr 536	. 2 |- ((x e. A /\ E!x(x e. A /\ ph)) -> (ph <-> U.{x e. A | ph} = x))
13		df-reu 1643	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
14	12, 13	sylan2b 452	1 |- ((x e. A /\ E!x e. A ph) -> (ph <-> U.{x e. A | ph} = x))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E!weu 1373  {cab 1456  E!wreu 1639  {crab 1640  U.cuni 2493

This theorem is referenced by:  reuuni2f 2873  reuuni4 2877  subadd 5343  divmul 5674  replimt 6692  cnid 8064  mulid 8069  hilid 8949

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-reu 1643  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-uni 2494

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