Theorem reuun1 3445

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 3326	. 2 |- A C_ ( A u. B )
2		orc 713	. . 3 |- ( ph -> ( ph \/ ps ) )
3	2	rgenw 2552	. 2 |- A. x e. A ( ph -> ( ph \/ ps ) )
4		reuss2 3443	. 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 709   A.wral 2475   E.wrex 2476   E!wreu 2477   u. cun 3155   C_ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-v 2765  df-un 3161  df-in 3163  df-ss 3170

This theorem is referenced by: (None)