MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  moaneu Unicode version

Theorem moaneu 2202
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 2183 . . 3  |-  ( E! x ph  ->  E* x ph )
2 nfeu1 2153 . . . 4  |-  F/ x E! x ph
32moanim 2199 . . 3  |-  ( E* x ( E! x ph  /\  ph )  <->  ( E! x ph  ->  E* x ph ) )
41, 3mpbir 200 . 2  |-  E* x
( E! x ph  /\ 
ph )
5 ancom 437 . . 3  |-  ( (
ph  /\  E! x ph )  <->  ( E! x ph  /\  ph ) )
65mobii 2179 . 2  |-  ( E* x ( ph  /\  E! x ph )  <->  E* x
( E! x ph  /\ 
ph ) )
74, 6mpbir 200 1  |-  E* x
( ph  /\  E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E!weu 2143   E*wmo 2144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148
  Copyright terms: Public domain W3C validator