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Theorem eupickbi 2358
Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
Assertion
Ref Expression
eupickbi  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )

Proof of Theorem eupickbi
StepHypRef Expression
1 eupicka 2356 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  A. x
( ph  ->  ps )
)
21ex 432 . 2  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  ->  A. x
( ph  ->  ps )
) )
3 euex 2310 . . 3  |-  ( E! x ph  ->  E. x ph )
4 exintr 1707 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x
( ph  /\  ps )
) )
53, 4syl5com 30 . 2  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  ->  E. x
( ph  /\  ps )
) )
62, 5impbid 191 1  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396   E.wex 1617   E!weu 2284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622  df-eu 2288  df-mo 2289
This theorem is referenced by:  sbaniota  31586
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