Theorem reuun1 3418

Description: Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)

Assertion

Ref	Expression
reuun1	|- ( ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) ) -> E! x e. A ph )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem reuun1

Step	Hyp	Ref	Expression
1		ssun1 3299	. 2 |- A C_ ( A u. B )
2		orc 712	. . 3 |- ( ph -> ( ph \/ ps ) )
3	2	rgenw 2532	. 2 |- A. x e. A ( ph -> ( ph \/ ps ) )
4		reuss2 3416	. 2 |- ( ( ( A C_ ( A u. B ) /\ A. x e. A ( ph -> ( ph \/ ps ) ) ) /\ ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) ) ) -> E! x e. A ph )
5	1, 3, 4	mpanl12 436	1 |- ( ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) ) -> E! x e. A ph )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 708   A.wral 2455   E.wrex 2456   E!wreu 2457   u. cun 3128   C_ wss 3130

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2740  df-un 3134  df-in 3136  df-ss 3143

This theorem is referenced by: (None)