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Theorem eupickbi 2346
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 2344 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  A. x
( ph  ->  ps )
)
21ex 424 . 2  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  ->  A. x
( ph  ->  ps )
) )
3 nfa1 1806 . . . . 5  |-  F/ x A. x ( ph  ->  ps )
4 ancl 530 . . . . . . 7  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ( ph  /\ 
ps ) ) )
5 simpl 444 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ph )
64, 5impbid1 195 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  <->  ( ph  /\  ps ) ) )
76sps 1770 . . . . 5  |-  ( A. x ( ph  ->  ps )  ->  ( ph  <->  (
ph  /\  ps )
) )
83, 7eubid 2287 . . . 4  |-  ( A. x ( ph  ->  ps )  ->  ( E! x ph  <->  E! x ( ph  /\ 
ps ) ) )
9 euex 2303 . . . 4  |-  ( E! x ( ph  /\  ps )  ->  E. x
( ph  /\  ps )
)
108, 9syl6bi 220 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E! x ph  ->  E. x
( ph  /\  ps )
) )
1110com12 29 . 2  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  ->  E. x
( ph  /\  ps )
) )
122, 11impbid 184 1  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550   E!weu 2280
This theorem is referenced by:  sbaniota  27550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285
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