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Theorem mopick2 2363
Description: "At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1651. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
mopick2  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps )  /\  E. x
( ph  /\  ch )
)  ->  E. x
( ph  /\  ps  /\  ch ) )

Proof of Theorem mopick2
StepHypRef Expression
1 nfmo1 2282 . . . 4  |-  F/ x E* x ph
2 nfe1 1784 . . . 4  |-  F/ x E. x ( ph  /\  ps )
31, 2nfan 1870 . . 3  |-  F/ x
( E* x ph  /\ 
E. x ( ph  /\ 
ps ) )
4 mopick 2354 . . . . . 6  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
54ancld 553 . . . . 5  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ( ph  /\  ps ) ) )
65anim1d 564 . . . 4  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  (
( ph  /\  ch )  ->  ( ( ph  /\  ps )  /\  ch )
) )
7 df-3an 970 . . . 4  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
86, 7syl6ibr 227 . . 3  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  (
( ph  /\  ch )  ->  ( ph  /\  ps  /\ 
ch ) ) )
93, 8eximd 1825 . 2  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( E. x ( ph  /\  ch )  ->  E. x
( ph  /\  ps  /\  ch ) ) )
1093impia 1188 1  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps )  /\  E. x
( ph  /\  ch )
)  ->  E. x
( ph  /\  ps  /\  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968   E.wex 1591   E*wmo 2269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 970  df-ex 1592  df-nf 1595  df-eu 2272  df-mo 2273
This theorem is referenced by: (None)
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