Theorem reuun1 3246

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

Assertion

Ref	Expression
reuun1	|- ( ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) ) -> E! x e. A ph )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem reuun1

Step	Hyp	Ref	Expression
1		ssun1 3135	. 2 |- A C_ ( A u. B )
2		orc 665	. . 3 |- ( ph -> ( ph \/ ps ) )
3	2	rgenw 2418	. 2 |- A. x e. A ( ph -> ( ph \/ ps ) )
4		reuss2 3244	. 2 |- ( ( ( A C_ ( A u. B ) /\ A. x e. A ( ph -> ( ph \/ ps ) ) ) /\ ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) ) ) -> E! x e. A ph )
5	1, 3, 4	mpanl12 426	1 |- ( ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) ) -> E! x e. A ph )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   A.wral 2348   E.wrex 2349   E!wreu 2350   u. cun 2971   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-v 2603  df-un 2977  df-in 2979  df-ss 2986

This theorem is referenced by: (None)