Theorem moaneu 2082

Description: Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.)

Assertion

Ref	Expression
moaneu	|- E* x ( ph /\ E! x ph )

Proof of Theorem moaneu

Step	Hyp	Ref	Expression
1		eumo 2038	. . 3 |- ( E! x ph -> E* x ph )
2		nfeu1 2017	. . . 4 |- F/ x E! x ph
3	2	moanim 2080	. . 3 |- ( E* x ( E! x ph /\ ph ) <-> ( E! x ph -> E* x ph ) )
4	1, 3	mpbir 145	. 2 |- E* x ( E! x ph /\ ph )
5		ancom 264	. . 3 |- ( ( ph /\ E! x ph ) <-> ( E! x ph /\ ph ) )
6	5	mobii 2043	. 2 |- ( E* x ( ph /\ E! x ph ) <-> E* x ( E! x ph /\ ph ) )
7	4, 6	mpbir 145	1 |- E* x ( ph /\ E! x ph )

Syntax hints:   -> wi 4   /\ wa 103   E!weu 2006   E*wmo 2007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010

This theorem is referenced by: (None)