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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
StepHypRef Expression
1 eumo 2038 . . 3  |-  ( E! x ph  ->  E* x ph )
2 nfeu1 2017 . . . 4  |-  F/ x E! x ph
32moanim 2080 . . 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 2043 . 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 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)
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