Theorem reuun2 2278

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 1650	. . . 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 1437	. . 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 1651	. . 3 |- (E!x e. (A u. B)ph <-> E!x(x e. (A u. B) /\ ph))
6		elun 2173	. . . . . 6 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
7	6	anbi1i 481	. . . . 5 |- ((x e. (A u. B) /\ ph) <-> ((x e. A \/ x e. B) /\ ph))
8		andir 605	. . . . 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 1387	. . 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 1651	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
14	4, 12, 13	3bitr4g 555	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 958  E.wex 980  E!weu 1380  E.wrex 1646  E!wreu 1647   u. cun 2045

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-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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1650  df-reu 1651  df-v 1812  df-un 2050

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