Theorem moaneu 2121

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 2077	. . 3 |- ( E! x ph -> E* x ph )
2		nfeu1 2056	. . . 4 |- F/ x E! x ph
3	2	moanim 2119	. . 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 2082	. 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 2045   E*wmo 2046

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

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049

This theorem is referenced by: (None)