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Theorem mopick2 2058
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 1593. (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 hbmo1 2013 . . . 4  |-  ( E* x ph  ->  A. x E* x ph )
2 hbe1 1454 . . . 4  |-  ( E. x ( ph  /\  ps )  ->  A. x E. x ( ph  /\  ps ) )
31, 2hban 1509 . . 3  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  A. x
( E* x ph  /\ 
E. x ( ph  /\ 
ps ) ) )
4 mopick 2053 . . . . . 6  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
54ancld 321 . . . . 5  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ( ph  /\  ps ) ) )
65anim1d 332 . . . 4  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  (
( ph  /\  ch )  ->  ( ( ph  /\  ps )  /\  ch )
) )
7 df-3an 947 . . . 4  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
86, 7syl6ibr 161 . . 3  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  (
( ph  /\  ch )  ->  ( ph  /\  ps  /\ 
ch ) ) )
93, 8eximdh 1573 . 2  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( E. x ( ph  /\  ch )  ->  E. x
( ph  /\  ps  /\  ch ) ) )
1093impia 1161 1  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps )  /\  E. x
( ph  /\  ch )
)  ->  E. x
( ph  /\  ps  /\  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945   E.wex 1451   E*wmo 1976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-3an 947  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979
This theorem is referenced by: (None)
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