Theorem reuss 3402

Description: Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)

Assertion

Ref	Expression
reuss	|- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> E! x e. A ph )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem reuss

Step	Hyp	Ref	Expression
1		idd 21	. . . 4 |- ( x e. A -> ( ph -> ph ) )
2	1	rgen 2518	. . 3 |- A. x e. A ( ph -> ph )
3		reuss2 3401	. . 3 |- ( ( ( A C_ B /\ A. x e. A ( ph -> ph ) ) /\ ( E. x e. A ph /\ E! x e. B ph ) ) -> E! x e. A ph )
4	2, 3	mpanl2 432	. 2 |- ( ( A C_ B /\ ( E. x e. A ph /\ E! x e. B ph ) ) -> E! x e. A ph )
5	4	3impb 1189	1 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> E! x e. A ph )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   e. wcel 2136   A.wral 2443   E.wrex 2444   E!wreu 2445   C_ wss 3115

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-3an 970  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2448  df-rex 2449  df-reu 2450  df-in 3121  df-ss 3128

This theorem is referenced by:  riotass  5824