Theorem reuuni3 2886

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 2885	. . 3 |- (E!x e. A ph -> U.{x e. A | ph} e. A)
2		reuuni3.1	. . . . 5 |- (x = y -> (ph <-> ps))
3	2	cbvrabv 1911	. . . 4 |- {x e. A | ph} = {y e. A | ps}
4	3	unieqi 2511	. . 3 |- U.{x e. A | ph} = U.{y e. A | ps}
5	1, 4	syl5eqelr 1553	. 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 2884	. . 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 705	1 |- (E!x e. A ph -> ch)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E!wreu 1647  {crab 1648  U.cuni 2503

This theorem is referenced by:  lble 6047  uzwo3lem2 6217  flleltt 6228

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-reu 1651  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-uni 2504

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