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Theorem eupickbi 2082
Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
eupickbi  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )

Proof of Theorem eupickbi
StepHypRef Expression
1 eupicka 2080 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  A. x
( ph  ->  ps )
)
21ex 114 . 2  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  ->  A. x
( ph  ->  ps )
) )
3 hba1 1521 . . . . 5  |-  ( A. x ( ph  ->  ps )  ->  A. x A. x ( ph  ->  ps ) )
4 ancl 316 . . . . . . 7  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ( ph  /\ 
ps ) ) )
5 simpl 108 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ph )
64, 5impbid1 141 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  <->  ( ph  /\  ps ) ) )
76sps 1518 . . . . 5  |-  ( A. x ( ph  ->  ps )  ->  ( ph  <->  (
ph  /\  ps )
) )
83, 7eubidh 2006 . . . 4  |-  ( A. x ( ph  ->  ps )  ->  ( E! x ph  <->  E! x ( ph  /\ 
ps ) ) )
9 euex 2030 . . . 4  |-  ( E! x ( ph  /\  ps )  ->  E. x
( ph  /\  ps )
)
108, 9syl6bi 162 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E! x ph  ->  E. x
( ph  /\  ps )
) )
1110com12 30 . 2  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  ->  E. x
( ph  /\  ps )
) )
122, 11impbid 128 1  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1330   E.wex 1469   E!weu 2000
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004
This theorem is referenced by: (None)
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