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Theorem eupickbi 2209
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 2207 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  A. x
( ph  ->  ps )
)
21ex 423 . 2  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  ->  A. x
( ph  ->  ps )
) )
3 nfa1 1756 . . . . 5  |-  F/ x A. x ( ph  ->  ps )
4 ancl 529 . . . . . . 7  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ( ph  /\ 
ps ) ) )
5 simpl 443 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ph )
64, 5impbid1 194 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  <->  ( ph  /\  ps ) ) )
76sps 1739 . . . . 5  |-  ( A. x ( ph  ->  ps )  ->  ( ph  <->  (
ph  /\  ps )
) )
83, 7eubid 2150 . . . 4  |-  ( A. x ( ph  ->  ps )  ->  ( E! x ph  <->  E! x ( ph  /\ 
ps ) ) )
9 euex 2166 . . . 4  |-  ( E! x ( ph  /\  ps )  ->  E. x
( ph  /\  ps )
)
108, 9syl6bi 219 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E! x ph  ->  E. x
( ph  /\  ps )
) )
1110com12 27 . 2  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  ->  E. x
( ph  /\  ps )
) )
122, 11impbid 183 1  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528   E!weu 2143
This theorem is referenced by:  sbaniota  27047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148
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