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Theorem eupickbi 2030
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 2028 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  A. x
( ph  ->  ps )
)
21ex 113 . 2  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  ->  A. x
( ph  ->  ps )
) )
3 hba1 1478 . . . . 5  |-  ( A. x ( ph  ->  ps )  ->  A. x A. x ( ph  ->  ps ) )
4 ancl 311 . . . . . . 7  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ( ph  /\ 
ps ) ) )
5 simpl 107 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ph )
64, 5impbid1 140 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  <->  ( ph  /\  ps ) ) )
76sps 1475 . . . . 5  |-  ( A. x ( ph  ->  ps )  ->  ( ph  <->  (
ph  /\  ps )
) )
83, 7eubidh 1954 . . . 4  |-  ( A. x ( ph  ->  ps )  ->  ( E! x ph  <->  E! x ( ph  /\ 
ps ) ) )
9 euex 1978 . . . 4  |-  ( E! x ( ph  /\  ps )  ->  E. x
( ph  /\  ps )
)
108, 9syl6bi 161 . . 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 127 1  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287   E.wex 1426   E!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by: (None)
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