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Theorem mopick 2131
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 1548 . . . 4  |-  ( (
ph  /\  ps )  ->  A. y ( ph  /\ 
ps ) )
2 hbs1 1965 . . . . 5  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
3 hbs1 1965 . . . . 5  |-  ( [ y  /  x ] ps  ->  A. x [ y  /  x ] ps )
42, 3hban 1569 . . . 4  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps )  ->  A. x ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
5 sbequ12 1793 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
6 sbequ12 1793 . . . . 5  |-  ( x  =  y  ->  ( ps 
<->  [ y  /  x ] ps ) )
75, 6anbi12d 473 . . . 4  |-  ( x  =  y  ->  (
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) ) )
81, 4, 7cbvexh 1777 . . 3  |-  ( E. x ( ph  /\  ps )  <->  E. y ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
9 ax-17 1548 . . . . . . 7  |-  ( ph  ->  A. y ph )
109mo3h 2106 . . . . . 6  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
11 ax-4 1532 . . . . . . 7  |-  ( A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  -> 
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
1211sps 1559 . . . . . 6  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
1310, 12sylbi 121 . . . . 5  |-  ( E* x ph  ->  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
14 sbequ2 1791 . . . . . . . . 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 258 . . . . . . 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 124 . . . . 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 1620 . . 3  |-  ( E. y ( [ y  /  x ] ph  /\ 
[ y  /  x ] ps )  ->  ( E* x ph  ->  ( ph  ->  ps ) ) )
218, 20sylbi 121 . 2  |-  ( E. x ( ph  /\  ps )  ->  ( E* x ph  ->  ( ph  ->  ps ) ) )
2221impcom 125 1  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1370   E.wex 1514   [wsb 1784   E*wmo 2054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557
This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057
This theorem is referenced by:  eupick  2132  mopick2  2136  moexexdc  2137  euexex  2138  morex  2956  imadif  5353  funimaexglem  5356
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