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Theorem moaneu 2051
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
StepHypRef Expression
1 eumo 2007 . . 3  |-  ( E! x ph  ->  E* x ph )
2 nfeu1 1986 . . . 4  |-  F/ x E! x ph
32moanim 2049 . . 3  |-  ( E* x ( E! x ph  /\  ph )  <->  ( E! x ph  ->  E* x ph ) )
41, 3mpbir 145 . 2  |-  E* x
( E! x ph  /\ 
ph )
5 ancom 264 . . 3  |-  ( (
ph  /\  E! x ph )  <->  ( E! x ph  /\  ph ) )
65mobii 2012 . 2  |-  ( E* x ( ph  /\  E! x ph )  <->  E* x
( E! x ph  /\ 
ph ) )
74, 6mpbir 145 1  |-  E* x
( ph  /\  E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E!weu 1975   E*wmo 1976
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979
This theorem is referenced by: (None)
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