Theorem moaneu 2163

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 2118	. . 3 |- ( E! x ph -> E* x ph )
2		nfeu1 2097	. . . 4 |- F/ x E! x ph
3	2	moanim 2161	. . 3 |- ( E* x ( E! x ph /\ ph ) <-> ( E! x ph -> E* x ph ) )
4	1, 3	mpbir 146	. 2 |- E* x ( E! x ph /\ ph )
5		ancom 266	. . 3 |- ( ( ph /\ E! x ph ) <-> ( E! x ph /\ ph ) )
6	5	mobii 2123	. 2 |- ( E* x ( ph /\ E! x ph ) <-> E* x ( E! x ph /\ ph ) )
7	4, 6	mpbir 146	1 |- E* x ( ph /\ E! x ph )

Syntax hints:   -> wi 4   /\ wa 104   E!weu 2086   E*wmo 2087

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090

This theorem is referenced by: (None)