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Theorem mopick2 2355
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 1647. (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 2274 . . . 4  |-  F/ x E* x ph
2 nfe1 1780 . . . 4  |-  F/ x E. x ( ph  /\  ps )
31, 2nfan 1863 . . 3  |-  F/ x
( E* x ph  /\ 
E. x ( ph  /\ 
ps ) )
4 mopick 2346 . . . . . 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 967 . . . 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 1818 . 2  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( E. x ( ph  /\  ch )  ->  E. x
( ph  /\  ps  /\  ch ) ) )
1093impia 1185 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 965   E.wex 1587   E*wmo 2261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-ex 1588  df-nf 1591  df-eu 2264  df-mo 2265
This theorem is referenced by: (None)
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