Theorem reuun2 3464

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 2492	. . 3 |- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
2		euor2 2114	. . 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 680	. 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 2493	. . 3 |- ( E! x e. ( A u. B ) ph <-> E! x ( x e. ( A u. B ) /\ ph ) )
5		elun 3322	. . . . . 6 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
6	5	anbi1i 458	. . . . 5 |- ( ( x e. ( A u. B ) /\ ph ) <-> ( ( x e. A \/ x e. B ) /\ ph ) )
7		andir 821	. . . . . 6 |- ( ( ( x e. A \/ x e. B ) /\ ph ) <-> ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) )
8		orcom 730	. . . . . 6 |- ( ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) <-> ( ( x e. B /\ ph ) \/ ( x e. A /\ ph ) ) )
9	7, 8	bitri 184	. . . . 5 |- ( ( ( x e. A \/ x e. B ) /\ ph ) <-> ( ( x e. B /\ ph ) \/ ( x e. A /\ ph ) ) )
10	6, 9	bitri 184	. . . 4 |- ( ( x e. ( A u. B ) /\ ph ) <-> ( ( x e. B /\ ph ) \/ ( x e. A /\ ph ) ) )
11	10	eubii 2064	. . 3 |- ( E! x ( x e. ( A u. B ) /\ ph ) <-> E! x ( ( x e. B /\ ph ) \/ ( x e. A /\ ph ) ) )
12	4, 11	bitri 184	. 2 |- ( E! x e. ( A u. B ) ph <-> E! x ( ( x e. B /\ ph ) \/ ( x e. A /\ ph ) ) )
13		df-reu 2493	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
14	3, 12, 13	3bitr4g 223	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 104   <-> wb 105   \/ wo 710   E.wex 1516   E!weu 2055   e. wcel 2178   E.wrex 2487   E!wreu 2488   u. cun 3172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-reu 2493  df-v 2778  df-un 3178

This theorem is referenced by: (None)