Theorem reuun2 2249

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 1626	. . . 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 1414	. . 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 1627	. . 3 |- (E!x e. (A u. B)ph <-> E!x(x e. (A u. B) /\ ph))
6		elun 2144	. . . . . 6 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
7	6	anbi1i 480	. . . . 5 |- ((x e. (A u. B) /\ ph) <-> ((x e. A \/ x e. B) /\ ph))
8		andir 603	. . . . 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 1364	. . 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 1627	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
14	4, 12, 13	3bitr4g 553	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.wex 956   e. wcel 1105  E!weu 1357  E.wrex 1622  E!wreu 1623   u. cun 2016

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-rex 1626  df-reu 1627  df-v 1787  df-un 2021

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