Theorem reuun2 3306

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 2381	. . 3 |- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
2		euor2 2018	. . 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 644	. 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 2382	. . 3 |- ( E! x e. ( A u. B ) ph <-> E! x ( x e. ( A u. B ) /\ ph ) )
5		elun 3164	. . . . . 6 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
6	5	anbi1i 449	. . . . 5 |- ( ( x e. ( A u. B ) /\ ph ) <-> ( ( x e. A \/ x e. B ) /\ ph ) )
7		andir 774	. . . . . 6 |- ( ( ( x e. A \/ x e. B ) /\ ph ) <-> ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) )
8		orcom 688	. . . . . 6 |- ( ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) <-> ( ( x e. B /\ ph ) \/ ( x e. A /\ ph ) ) )
9	7, 8	bitri 183	. . . . 5 |- ( ( ( x e. A \/ x e. B ) /\ ph ) <-> ( ( x e. B /\ ph ) \/ ( x e. A /\ ph ) ) )
10	6, 9	bitri 183	. . . 4 |- ( ( x e. ( A u. B ) /\ ph ) <-> ( ( x e. B /\ ph ) \/ ( x e. A /\ ph ) ) )
11	10	eubii 1969	. . 3 |- ( E! x ( x e. ( A u. B ) /\ ph ) <-> E! x ( ( x e. B /\ ph ) \/ ( x e. A /\ ph ) ) )
12	4, 11	bitri 183	. 2 |- ( E! x e. ( A u. B ) ph <-> E! x ( ( x e. B /\ ph ) \/ ( x e. A /\ ph ) ) )
13		df-reu 2382	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
14	3, 12, 13	3bitr4g 222	1 |- ( -. E. x e. B ph -> ( E! x e. ( A u. B ) ph <-> E! x e. A ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 670   E.wex 1436   e. wcel 1448   E!weu 1960   E.wrex 2376   E!wreu 2377   u. cun 3019

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-reu 2382  df-v 2643  df-un 3025

This theorem is referenced by: (None)