Theorem reuun1 3404

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 3285	. 2 |- A C_ ( A u. B )
2		orc 702	. . 3 |- ( ph -> ( ph \/ ps ) )
3	2	rgenw 2521	. 2 |- A. x e. A ( ph -> ( ph \/ ps ) )
4		reuss2 3402	. 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 433	1 |- ( ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) ) -> E! x e. A ph )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   A.wral 2444   E.wrex 2445   E!wreu 2446   u. cun 3114   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-v 2728  df-un 3120  df-in 3122  df-ss 3129

This theorem is referenced by: (None)