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Theorem mopick 2055
Description: "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)
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 ax-17 1491 . . . 4  |-  ( (
ph  /\  ps )  ->  A. y ( ph  /\ 
ps ) )
2 hbs1 1891 . . . . 5  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
3 hbs1 1891 . . . . 5  |-  ( [ y  /  x ] ps  ->  A. x [ y  /  x ] ps )
42, 3hban 1511 . . . 4  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps )  ->  A. x ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
5 sbequ12 1729 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
6 sbequ12 1729 . . . . 5  |-  ( x  =  y  ->  ( ps 
<->  [ y  /  x ] ps ) )
75, 6anbi12d 464 . . . 4  |-  ( x  =  y  ->  (
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) ) )
81, 4, 7cbvexh 1713 . . 3  |-  ( E. x ( ph  /\  ps )  <->  E. y ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
9 ax-17 1491 . . . . . . 7  |-  ( ph  ->  A. y ph )
109mo3h 2030 . . . . . 6  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
11 ax-4 1472 . . . . . . 7  |-  ( A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  -> 
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
1211sps 1502 . . . . . 6  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
1310, 12sylbi 120 . . . . 5  |-  ( E* x ph  ->  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
14 sbequ2 1727 . . . . . . . . 9  |-  ( x  =  y  ->  ( [ y  /  x ] ps  ->  ps )
)
1514imim2i 12 . . . . . . . 8  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  ->  (
( ph  /\  [ y  /  x ] ph )  ->  ( [ y  /  x ] ps  ->  ps ) ) )
1615expd 256 . . . . . . 7  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  ->  ( ph  ->  ( [ y  /  x ] ph  ->  ( [ y  /  x ] ps  ->  ps ) ) ) )
1716com4t 85 . . . . . 6  |-  ( [ y  /  x ] ph  ->  ( [ y  /  x ] ps  ->  ( ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  -> 
( ph  ->  ps )
) ) )
1817imp 123 . . . . 5  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps )  ->  ( ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( ph  ->  ps ) ) )
1913, 18syl5 32 . . . 4  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps )  ->  ( E* x ph  ->  ( ph  ->  ps ) ) )
2019exlimiv 1562 . . 3  |-  ( E. y ( [ y  /  x ] ph  /\ 
[ y  /  x ] ps )  ->  ( E* x ph  ->  ( ph  ->  ps ) ) )
218, 20sylbi 120 . 2  |-  ( E. x ( ph  /\  ps )  ->  ( E* x ph  ->  ( ph  ->  ps ) ) )
2221impcom 124 1  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1314   E.wex 1453   [wsb 1720   E*wmo 1978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981
This theorem is referenced by:  eupick  2056  mopick2  2060  moexexdc  2061  euexex  2062  morex  2841  imadif  5173  funimaexglem  5176
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