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Theorem mopick 2334
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 2276 . . 3  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
2 sp 1914 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( ph  ->  x  =  y ) )
3 pm3.45 842 . . . . . . 7  |-  ( (
ph  ->  x  =  y )  ->  ( ( ph  /\  ps )  -> 
( x  =  y  /\  ps ) ) )
43aleximi 1698 . . . . . 6  |-  ( A. x ( ph  ->  x  =  y )  -> 
( E. x (
ph  /\  ps )  ->  E. x ( x  =  y  /\  ps ) ) )
5 sb56 2227 . . . . . . 7  |-  ( E. x ( x  =  y  /\  ps )  <->  A. x ( x  =  y  ->  ps )
)
6 sp 1914 . . . . . . 7  |-  ( A. x ( x  =  y  ->  ps )  ->  ( x  =  y  ->  ps ) )
75, 6sylbi 198 . . . . . 6  |-  ( E. x ( x  =  y  /\  ps )  ->  ( x  =  y  ->  ps ) )
84, 7syl6 34 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( E. x (
ph  /\  ps )  ->  ( x  =  y  ->  ps ) ) )
92, 8syl5d 69 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  -> 
( E. x (
ph  /\  ps )  ->  ( ph  ->  ps ) ) )
109exlimiv 1770 . . 3  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  ( E. x ( ph  /\  ps )  ->  ( ph  ->  ps ) ) )
111, 10sylbi 198 . 2  |-  ( E* x ph  ->  ( E. x ( ph  /\  ps )  ->  ( ph  ->  ps ) ) )
1211imp 430 1  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370   A.wal 1435   E.wex 1657   E*wmo 2270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662  df-eu 2273  df-mo 2274
This theorem is referenced by:  eupick  2335  mopick2  2339  moexex  2340  morex  3254  imadif  5676  cmetss  22282
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