Theorem reuss 3280

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 2428	. . 3 |- A. x e. A ( ph -> ph )
3		reuss2 3279	. . 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 426	. 2 |- ( ( A C_ B /\ ( E. x e. A ph /\ E! x e. B ph ) ) -> E! x e. A ph )
5	4	3impb 1139	1 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> E! x e. A ph )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 924   e. wcel 1438   A.wral 2359   E.wrex 2360   E!wreu 2361   C_ wss 2999

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-ral 2364  df-rex 2365  df-reu 2366  df-in 3005  df-ss 3012

This theorem is referenced by:  riotass  5635