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Theorem mopick 2361
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 2282 . . 3  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
2 sp 1808 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( ph  ->  x  =  y ) )
3 id 22 . . . . . . . 8  |-  ( (
ph  ->  x  =  y )  ->  ( ph  ->  x  =  y ) )
43anim1d 564 . . . . . . 7  |-  ( (
ph  ->  x  =  y )  ->  ( ( ph  /\  ps )  -> 
( x  =  y  /\  ps ) ) )
54aleximi 1632 . . . . . 6  |-  ( A. x ( ph  ->  x  =  y )  -> 
( E. x (
ph  /\  ps )  ->  E. x ( x  =  y  /\  ps ) ) )
6 sb56 2154 . . . . . . 7  |-  ( E. x ( x  =  y  /\  ps )  <->  A. x ( x  =  y  ->  ps )
)
7 sp 1808 . . . . . . 7  |-  ( A. x ( x  =  y  ->  ps )  ->  ( x  =  y  ->  ps ) )
86, 7sylbi 195 . . . . . 6  |-  ( E. x ( x  =  y  /\  ps )  ->  ( x  =  y  ->  ps ) )
95, 8syl6 33 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( E. x (
ph  /\  ps )  ->  ( x  =  y  ->  ps ) ) )
102, 9syl5d 67 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  -> 
( E. x (
ph  /\  ps )  ->  ( ph  ->  ps ) ) )
1110exlimiv 1698 . . 3  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  ( E. x ( ph  /\  ps )  ->  ( ph  ->  ps ) ) )
121, 11sylbi 195 . 2  |-  ( E* x ph  ->  ( E. x ( ph  /\  ps )  ->  ( ph  ->  ps ) ) )
1312imp 429 1  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1377   E.wex 1596   E*wmo 2276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600  df-eu 2279  df-mo 2280
This theorem is referenced by:  eupick  2364  mopick2  2370  moexex  2371  moexexOLD  2372  morex  3287  imadif  5661  cmetss  21488
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