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

Theorem moanim 2212
Description: Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.)
Hypothesis
Ref Expression
moanim.1  |-  F/ x ph
Assertion
Ref Expression
moanim  |-  ( E* x ( ph  /\  ps )  <->  ( ph  ->  E* x ps ) )

Proof of Theorem moanim
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 impexp 433 . . . . 5  |-  ( ( ( ph  /\  ps )  ->  x  =  y )  <->  ( ph  ->  ( ps  ->  x  =  y ) ) )
21albii 1556 . . . 4  |-  ( A. x ( ( ph  /\ 
ps )  ->  x  =  y )  <->  A. x
( ph  ->  ( ps 
->  x  =  y
) ) )
3 moanim.1 . . . . 5  |-  F/ x ph
4319.21 1803 . . . 4  |-  ( A. x ( ph  ->  ( ps  ->  x  =  y ) )  <->  ( ph  ->  A. x ( ps 
->  x  =  y
) ) )
52, 4bitri 240 . . 3  |-  ( A. x ( ( ph  /\ 
ps )  ->  x  =  y )  <->  ( ph  ->  A. x ( ps 
->  x  =  y
) ) )
65exbii 1572 . 2  |-  ( E. y A. x ( ( ph  /\  ps )  ->  x  =  y )  <->  E. y ( ph  ->  A. x ( ps 
->  x  =  y
) ) )
7 nfv 1609 . . 3  |-  F/ y ( ph  /\  ps )
87mo2 2185 . 2  |-  ( E* x ( ph  /\  ps )  <->  E. y A. x
( ( ph  /\  ps )  ->  x  =  y ) )
9 nfv 1609 . . . . 5  |-  F/ y ps
109mo2 2185 . . . 4  |-  ( E* x ps  <->  E. y A. x ( ps  ->  x  =  y ) )
1110imbi2i 303 . . 3  |-  ( (
ph  ->  E* x ps )  <->  ( ph  ->  E. y A. x ( ps  ->  x  =  y ) ) )
12 19.37v 1852 . . 3  |-  ( E. y ( ph  ->  A. x ( ps  ->  x  =  y ) )  <-> 
( ph  ->  E. y A. x ( ps  ->  x  =  y ) ) )
1311, 12bitr4i 243 . 2  |-  ( (
ph  ->  E* x ps )  <->  E. y ( ph  ->  A. x ( ps 
->  x  =  y
) ) )
146, 8, 133bitr4i 268 1  |-  ( E* x ( ph  /\  ps )  <->  ( ph  ->  E* x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531   F/wnf 1534   E*wmo 2157
This theorem is referenced by:  moanimv  2214  moaneu  2215  moanmo  2216  2eu1  2236
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161
  Copyright terms: Public domain W3C validator