Theorem reuun2 2281

Description: Transfer uniqueness to a smaller or larger class.

Assertion

Ref	Expression
reuun2	|- (-. E.x e. B ph -> (E!x e. (A u. B)ph <-> E!x e. A ph))

Distinct variable groups:   x,A   x,B

Proof of Theorem reuun2

Step	Hyp	Ref	Expression
1		df-rex 1653	. . . 4 |- (E.x e. B ph <-> E.x(x e. B /\ ph))
2	1	negbii 187	. . 3 |- (-. E.x e. B ph <-> -. E.x(x e. B /\ ph))
3		euor2 1440	. . 3 |- (-. E.x(x e. B /\ ph) -> (E!x((x e. B /\ ph) \/ (x e. A /\ ph)) <-> E!x(x e. A /\ ph)))
4	2, 3	sylbi 199	. 2 |- (-. E.x e. B ph -> (E!x((x e. B /\ ph) \/ (x e. A /\ ph)) <-> E!x(x e. A /\ ph)))
5		df-reu 1654	. . 3 |- (E!x e. (A u. B)ph <-> E!x(x e. (A u. B) /\ ph))
6		elun 2176	. . . . . 6 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
7	6	anbi1i 483	. . . . 5 |- ((x e. (A u. B) /\ ph) <-> ((x e. A \/ x e. B) /\ ph))
8		andir 607	. . . . 5 |- (((x e. A \/ x e. B) /\ ph) <-> ((x e. A /\ ph) \/ (x e. B /\ ph)))
9		orcom 246	. . . . 5 |- (((x e. A /\ ph) \/ (x e. B /\ ph)) <-> ((x e. B /\ ph) \/ (x e. A /\ ph)))
10	7, 8, 9	3bitr 177	. . . 4 |- ((x e. (A u. B) /\ ph) <-> ((x e. B /\ ph) \/ (x e. A /\ ph)))
11	10	eubii 1389	. . 3 |- (E!x(x e. (A u. B) /\ ph) <-> E!x((x e. B /\ ph) \/ (x e. A /\ ph)))
12	5, 11	bitr 173	. 2 |- (E!x e. (A u. B)ph <-> E!x((x e. B /\ ph) \/ (x e. A /\ ph)))
13		df-reu 1654	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
14	4, 12, 13	3bitr4g 557	1 |- (-. E.x e. B ph -> (E!x e. (A u. B)ph <-> E!x e. A ph))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   e. wcel 960  E.wex 982  E!weu 1382  E.wrex 1649  E!wreu 1650   u. cun 2048

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-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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-rex 1653  df-reu 1654  df-v 1815  df-un 2053

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