Theorem reuun2 3539

Description: Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)

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

Allowed substitution hint:   ph(x)

Proof of Theorem reuun2

Step	Hyp	Ref	Expression
1		df-rex 2621	. . 3 |- (E.x e. B ph <-> E.x(x e. B /\ ph))
2		euor2 2272	. . 3 |- (-. E.x(x e. B /\ ph) -> (E!x((x e. B /\ ph) \/ (x e. A /\ ph)) <-> E!x(x e. A /\ ph)))
3	1, 2	sylnbi 297	. 2 |- (-. E.x e. B ph -> (E!x((x e. B /\ ph) \/ (x e. A /\ ph)) <-> E!x(x e. A /\ ph)))
4		df-reu 2622	. . 3 |- (E!x e. (A u. B)ph <-> E!x(x e. (A u. B) /\ ph))
5		elun 3221	. . . . . 6 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
6	5	anbi1i 676	. . . . 5 |- ((x e. (A u. B) /\ ph) <-> ((x e. A \/ x e. B) /\ ph))
7		andir 838	. . . . . 6 |- (((x e. A \/ x e. B) /\ ph) <-> ((x e. A /\ ph) \/ (x e. B /\ ph)))
8		orcom 376	. . . . . 6 |- (((x e. A /\ ph) \/ (x e. B /\ ph)) <-> ((x e. B /\ ph) \/ (x e. A /\ ph)))
9	7, 8	bitri 240	. . . . 5 |- (((x e. A \/ x e. B) /\ ph) <-> ((x e. B /\ ph) \/ (x e. A /\ ph)))
10	6, 9	bitri 240	. . . 4 |- ((x e. (A u. B) /\ ph) <-> ((x e. B /\ ph) \/ (x e. A /\ ph)))
11	10	eubii 2213	. . 3 |- (E!x(x e. (A u. B) /\ ph) <-> E!x((x e. B /\ ph) \/ (x e. A /\ ph)))
12	4, 11	bitri 240	. 2 |- (E!x e. (A u. B)ph <-> E!x((x e. B /\ ph) \/ (x e. A /\ ph)))
13		df-reu 2622	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
14	3, 12, 13	3bitr4g 279	1 |- (-. E.x e. B ph -> (E!x e. (A u. B)ph <-> E!x e. A ph))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358  E.wex 1541   e. wcel 1710  E!weu 2204  E.wrex 2616  E!wreu 2617   u. cun 3208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-reu 2622  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215

This theorem is referenced by: (None)