Theorem moaneu 2130

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 2086	. . 3 |- ( E! x ph -> E* x ph )
2		nfeu1 2065	. . . 4 |- F/ x E! x ph
3	2	moanim 2128	. . 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 2091	. 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 2054   E*wmo 2055

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058

This theorem is referenced by: (None)