Theorem reuss 3490

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 2586	. . 3 |- A. x e. A ( ph -> ph )
3		reuss2 3489	. . 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 1226	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 1005   e. wcel 2202   A.wral 2511   E.wrex 2512   E!wreu 2513   C_ wss 3201

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2516  df-rex 2517  df-reu 2518  df-in 3207  df-ss 3214

This theorem is referenced by:  riotass  6011