Theorem reuuni3 2892

Description: Derive the property ch of "the unique element in A such that ph" when expressed explicitly as U.{y e. A | ps}.

Hypotheses

Ref	Expression
reuuni3.1	|- (x = y -> (ph <-> ps))
reuuni3.2	|- (x = U.{y e. A | ps} -> (ph <-> ch))

Assertion

Ref	Expression
reuuni3	|- (E!x e. A ph -> ch)

Distinct variable groups:   x,y,A   ph,y   ps,x   ch,x

Proof of Theorem reuuni3

Step	Hyp	Ref	Expression
1		reucl 2891	. . 3 |- (E!x e. A ph -> U.{x e. A | ph} e. A)
2		reuuni3.1	. . . . 5 |- (x = y -> (ph <-> ps))
3	2	cbvrabv 1914	. . . 4 |- {x e. A | ph} = {y e. A | ps}
4	3	unieqi 2515	. . 3 |- U.{x e. A | ph} = U.{y e. A | ps}
5	1, 4	syl5eqelr 1556	. 2 |- (E!x e. A ph -> U.{y e. A | ps} e. A)
6		reuuni3.2	. . . 4 |- (x = U.{y e. A | ps} -> (ph <-> ch))
7	6	reuuni2 2890	. . 3 |- ((U.{y e. A | ps} e. A /\ E!x e. A ph) -> (ch <-> U.{x e. A | ph} = U.{y e. A | ps}))
8	4, 7	mpbiri 194	. 2 |- ((U.{y e. A | ps} e. A /\ E!x e. A ph) -> ch)
9	5, 8	mpancom 707	1 |- (E!x e. A ph -> ch)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E!wreu 1650  {crab 1651  U.cuni 2507

This theorem is referenced by:  lble 6049  uzwo3lem2 6219  flleltt 6230

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-reu 1654  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-uni 2508

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