Theorem reuss 3453

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 2558	. . 3 |- A. x e. A ( ph -> ph )
3		reuss2 3452	. . 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 435	. 2 |- ( ( A C_ B /\ ( E. x e. A ph /\ E! x e. B ph ) ) -> E! x e. A ph )
5	4	3impb 1201	1 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> E! x e. A ph )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   e. wcel 2175   A.wral 2483   E.wrex 2484   E!wreu 2485   C_ wss 3165

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-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-ral 2488  df-rex 2489  df-reu 2490  df-in 3171  df-ss 3178

This theorem is referenced by:  riotass  5926