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Theorem mopick 2205
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 nfv 1605 . . . 4  |-  F/ y ( ph  /\  ps )
2 nfs1v 2045 . . . . 5  |-  F/ x [ y  /  x ] ph
3 nfs1v 2045 . . . . 5  |-  F/ x [ y  /  x ] ps
42, 3nfan 1771 . . . 4  |-  F/ x
( [ y  /  x ] ph  /\  [
y  /  x ] ps )
5 sbequ12 1860 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
6 sbequ12 1860 . . . . 5  |-  ( x  =  y  ->  ( ps 
<->  [ y  /  x ] ps ) )
75, 6anbi12d 691 . . . 4  |-  ( x  =  y  ->  (
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) ) )
81, 4, 7cbvex 1925 . . 3  |-  ( E. x ( ph  /\  ps )  <->  E. y ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
9 nfv 1605 . . . . . . 7  |-  F/ y
ph
109mo3 2174 . . . . . 6  |-  ( E* x ph  <->  A. x A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
11 sp 1716 . . . . . . 7  |-  ( A. y ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  -> 
( ( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
1211sps 1739 . . . . . 6  |-  ( A. x A. y ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y ) )
1310, 12sylbi 187 . . . . 5  |-  ( E* x ph  ->  (
( ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
14 sbequ2 1631 . . . . . . . . 9  |-  ( x  =  y  ->  ( [ y  /  x ] ps  ->  ps )
)
1514imim2i 13 . . . . . . . 8  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  ->  (
( ph  /\  [ y  /  x ] ph )  ->  ( [ y  /  x ] ps  ->  ps ) ) )
1615exp3a 425 . . . . . . 7  |-  ( ( ( ph  /\  [
y  /  x ] ph )  ->  x  =  y )  ->  ( ph  ->  ( [ y  /  x ] ph  ->  ( [ y  /  x ] ps  ->  ps ) ) ) )
1716com4t 79 . . . . . 6  |-  ( [ y  /  x ] ph  ->  ( [ y  /  x ] ps  ->  ( ( ( ph  /\ 
[ y  /  x ] ph )  ->  x  =  y )  -> 
( ph  ->  ps )
) ) )
1817imp 418 . . . . 5  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps )  ->  ( ( (
ph  /\  [ y  /  x ] ph )  ->  x  =  y )  ->  ( ph  ->  ps ) ) )
1913, 18syl5 28 . . . 4  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps )  ->  ( E* x ph  ->  ( ph  ->  ps ) ) )
2019exlimiv 1666 . . 3  |-  ( E. y ( [ y  /  x ] ph  /\ 
[ y  /  x ] ps )  ->  ( E* x ph  ->  ( ph  ->  ps ) ) )
218, 20sylbi 187 . 2  |-  ( E. x ( ph  /\  ps )  ->  ( E* x ph  ->  ( ph  ->  ps ) ) )
2221impcom 419 1  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623   [wsb 1629   E*wmo 2144
This theorem is referenced by:  eupick  2206  mopick2  2210  moexex  2212  morex  2949  imadif  5327  cmetss  18740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148
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