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Theorem moaneu 2342
Description: Nested "at most one" and uniqueness 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 2323 . . 3  |-  ( E! x ph  ->  E* x ph )
2 nfeu1 2293 . . . 4  |-  F/ x E! x ph
32moanim 2339 . . 3  |-  ( E* x ( E! x ph  /\  ph )  <->  ( E! x ph  ->  E* x ph ) )
41, 3mpbir 202 . 2  |-  E* x
( E! x ph  /\ 
ph )
5 ancom 439 . . 3  |-  ( (
ph  /\  E! x ph )  <->  ( E! x ph  /\  ph ) )
65mobii 2319 . 2  |-  ( E* x ( ph  /\  E! x ph )  <->  E* x
( E! x ph  /\ 
ph ) )
74, 6mpbir 202 1  |-  E* x
( ph  /\  E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E!weu 2283   E*wmo 2284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288
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