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Theorem mopick 2026
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 1464 . . . 4  |-  ( (
ph  /\  ps )  ->  A. y ( ph  /\ 
ps ) )
2 hbs1 1862 . . . . 5  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
3 hbs1 1862 . . . . 5  |-  ( [ y  /  x ] ps  ->  A. x [ y  /  x ] ps )
42, 3hban 1484 . . . 4  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps )  ->  A. x ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
5 sbequ12 1701 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
6 sbequ12 1701 . . . . 5  |-  ( x  =  y  ->  ( ps 
<->  [ y  /  x ] ps ) )
75, 6anbi12d 457 . . . 4  |-  ( x  =  y  ->  (
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) ) )
81, 4, 7cbvexh 1685 . . 3  |-  ( E. x ( ph  /\  ps )  <->  E. y ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
9 ax-17 1464 . . . . . . 7  |-  ( ph  ->  A. y ph )
109mo3h 2001 . . . . . 6  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
11 ax-4 1445 . . . . . . 7  |-  ( A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  -> 
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
1211sps 1475 . . . . . 6  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
1310, 12sylbi 119 . . . . 5  |-  ( E* x ph  ->  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
14 sbequ2 1699 . . . . . . . . 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 254 . . . . . . 7  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  ->  ( ph  ->  ( [ y  /  x ] ph  ->  ( [ y  /  x ] ps  ->  ps ) ) ) )
1716com4t 84 . . . . . 6  |-  ( [ y  /  x ] ph  ->  ( [ y  /  x ] ps  ->  ( ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  -> 
( ph  ->  ps )
) ) )
1817imp 122 . . . . 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 1534 . . 3  |-  ( E. y ( [ y  /  x ] ph  /\ 
[ y  /  x ] ps )  ->  ( E* x ph  ->  ( ph  ->  ps ) ) )
218, 20sylbi 119 . 2  |-  ( E. x ( ph  /\  ps )  ->  ( E* x ph  ->  ( ph  ->  ps ) ) )
2221impcom 123 1  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287   E.wex 1426   [wsb 1692   E*wmo 1949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by:  eupick  2027  mopick2  2031  moexexdc  2032  euexex  2033  morex  2797  imadif  5080  funimaexglem  5083
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