Theorem reuuni1 2877

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 2876	. . . . . . 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 515	. . 3 |- ((x e. A /\ E!x(x e. A /\ ph)) -> (ph <-> U.{x | (x e. A /\ ph)} = x))
9		df-rab 1649	. . . . 5 |- {x e. A | ph} = {x | (x e. A /\ ph)}
10	9	unieqi 2506	. . . 4 |- U.{x e. A | ph} = U.{x | (x e. A /\ ph)}
11	10	eqeq1i 1479	. . 3 |- (U.{x e. A | ph} = x <-> U.{x | (x e. A /\ ph)} = x)
12	8, 11	syl6bbr 537	. 2 |- ((x e. A /\ E!x(x e. A /\ ph)) -> (ph <-> U.{x e. A | ph} = x))
13		df-reu 1648	. 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 954   e. wcel 956  E!weu 1378  {cab 1461  E!wreu 1644  {crab 1645  U.cuni 2498

This theorem is referenced by:  reuuni2f 2878  reuuni4 2882  subadd 5351  divmul 5682  replimt 6700  cnid 8079  mulid 8084  hilid 8967

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-reu 1648  df-rab 1649  df-v 1808  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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