Theorem reuss 3462

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 2561	. . 3 |- A. x e. A ( ph -> ph )
3		reuss2 3461	. . 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 1202	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 981   e. wcel 2178   A.wral 2486   E.wrex 2487   E!wreu 2488   C_ wss 3174

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-ral 2491  df-rex 2492  df-reu 2493  df-in 3180  df-ss 3187

This theorem is referenced by:  riotass  5950