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

Theorem mopick 2307
Description: "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.)
Assertion
Ref Expression
mopick  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )

Proof of Theorem mopick
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 mo2v 2245 . . 3  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
2 sp 1883 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( ph  ->  x  =  y ) )
3 pm3.45 835 . . . . . . 7  |-  ( (
ph  ->  x  =  y )  ->  ( ( ph  /\  ps )  -> 
( x  =  y  /\  ps ) ) )
43aleximi 1674 . . . . . 6  |-  ( A. x ( ph  ->  x  =  y )  -> 
( E. x (
ph  /\  ps )  ->  E. x ( x  =  y  /\  ps ) ) )
5 sb56 2196 . . . . . . 7  |-  ( E. x ( x  =  y  /\  ps )  <->  A. x ( x  =  y  ->  ps )
)
6 sp 1883 . . . . . . 7  |-  ( A. x ( x  =  y  ->  ps )  ->  ( x  =  y  ->  ps ) )
75, 6sylbi 195 . . . . . 6  |-  ( E. x ( x  =  y  /\  ps )  ->  ( x  =  y  ->  ps ) )
84, 7syl6 31 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( E. x (
ph  /\  ps )  ->  ( x  =  y  ->  ps ) ) )
92, 8syl5d 66 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  -> 
( E. x (
ph  /\  ps )  ->  ( ph  ->  ps ) ) )
109exlimiv 1743 . . 3  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  ( E. x ( ph  /\  ps )  ->  ( ph  ->  ps ) ) )
111, 10sylbi 195 . 2  |-  ( E* x ph  ->  ( E. x ( ph  /\  ps )  ->  ( ph  ->  ps ) ) )
1211imp 427 1  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1403   E.wex 1633   E*wmo 2239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-12 1878  ax-13 2026
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1634  df-nf 1638  df-eu 2242  df-mo 2243
This theorem is referenced by:  eupick  2309  mopick2  2313  moexex  2314  morex  3232  imadif  5600  cmetss  21937
  Copyright terms: Public domain W3C validator