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Theorem mopick 1994
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 1435 . . . 4  |-  ( (
ph  /\  ps )  ->  A. y ( ph  /\ 
ps ) )
2 hbs1 1830 . . . . 5  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
3 hbs1 1830 . . . . 5  |-  ( [ y  /  x ] ps  ->  A. x [ y  /  x ] ps )
42, 3hban 1455 . . . 4  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps )  ->  A. x ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
5 sbequ12 1670 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
6 sbequ12 1670 . . . . 5  |-  ( x  =  y  ->  ( ps 
<->  [ y  /  x ] ps ) )
75, 6anbi12d 450 . . . 4  |-  ( x  =  y  ->  (
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) ) )
81, 4, 7cbvexh 1654 . . 3  |-  ( E. x ( ph  /\  ps )  <->  E. y ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
9 ax-17 1435 . . . . . . 7  |-  ( ph  ->  A. y ph )
109mo3h 1969 . . . . . 6  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
11 ax-4 1416 . . . . . . 7  |-  ( A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  -> 
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
1211sps 1446 . . . . . 6  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
1310, 12sylbi 118 . . . . 5  |-  ( E* x ph  ->  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
14 sbequ2 1668 . . . . . . . . 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 249 . . . . . . 7  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  ->  ( ph  ->  ( [ y  /  x ] ph  ->  ( [ y  /  x ] ps  ->  ps ) ) ) )
1716com4t 83 . . . . . 6  |-  ( [ y  /  x ] ph  ->  ( [ y  /  x ] ps  ->  ( ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  -> 
( ph  ->  ps )
) ) )
1817imp 119 . . . . 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 1505 . . 3  |-  ( E. y ( [ y  /  x ] ph  /\ 
[ y  /  x ] ps )  ->  ( E* x ph  ->  ( ph  ->  ps ) ) )
218, 20sylbi 118 . 2  |-  ( E. x ( ph  /\  ps )  ->  ( E* x ph  ->  ( ph  ->  ps ) ) )
2221impcom 120 1  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   A.wal 1257   E.wex 1397   [wsb 1661   E*wmo 1917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920
This theorem is referenced by:  eupick  1995  mopick2  1999  moexexdc  2000  euexex  2001  morex  2748  imadif  5007  funimaexglem  5010
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