Theorem reuuni1 1955

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 1954	. . . . . . . 8 |- (E!x(x e. A /\ ph) -> ((x e. A /\ ph) <-> U.{x | (x e. A /\ ph)} = x))
2	1	biimpd 135	. . . . . . 7 |- (E!x(x e. A /\ ph) -> ((x e. A /\ ph) -> U.{x | (x e. A /\ ph)} = x))
3	2	exp3a 292	. . . . . 6 |- (E!x(x e. A /\ ph) -> (x e. A -> (ph -> U.{x | (x e. A /\ ph)} = x)))
4	3	com12 13	. . . . 5 |- (x e. A -> (E!x(x e. A /\ ph) -> (ph -> U.{x | (x e. A /\ ph)} = x)))
5	4	imp 277	. . . 4 |- ((x e. A /\ E!x(x e. A /\ ph)) -> (ph -> U.{x | (x e. A /\ ph)} = x))
6		pm3.27 260	. . . . . 6 |- ((x e. A /\ ph) -> ph)
7	1, 6	syl6bir 188	. . . . 5 |- (E!x(x e. A /\ ph) -> (U.{x | (x e. A /\ ph)} = x -> ph))
8	7	adantl 305	. . . 4 |- ((x e. A /\ E!x(x e. A /\ ph)) -> (U.{x | (x e. A /\ ph)} = x -> ph))
9	5, 8	impbid 397	. . 3 |- ((x e. A /\ E!x(x e. A /\ ph)) -> (ph <-> U.{x | (x e. A /\ ph)} = x))
10		df-rab 1208	. . . . 5 |- {x e. A | ph} = {x | (x e. A /\ ph)}
11	10	unieqi 1928	. . . 4 |- U.{x e. A | ph} = U.{x | (x e. A /\ ph)}
12	11	cleq1i 1108	. . 3 |- (U.{x e. A | ph} = x <-> U.{x | (x e. A /\ ph)} = x)
13	9, 12	syl6bbr 416	. 2 |- ((x e. A /\ E!x(x e. A /\ ph)) -> (ph <-> U.{x e. A | ph} = x))
14		df-reu 1207	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
15	13, 14	sylan2b 347	1 |- ((x e. A /\ E!x e. A ph) -> (ph <-> U.{x e. A | ph} = x))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E!weu 1007  {cab 1090   = wceq 1091   e. wcel 1092  E!wreu 1203  {crab 1204  U.cuni 1919

This theorem is referenced by:  reuuni2 1956  reuuni4 1959  subadd 4143  divmul 4218  replimt 4798

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-reu 1207  df-rab 1208  df-v 1349  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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