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Theorem eupickbiOLD 2369
Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
eupickbiOLD  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )

Proof of Theorem eupickbiOLD
StepHypRef Expression
1 eupicka 2365 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  A. x
( ph  ->  ps )
)
21ex 434 . 2  |-  ( E! x ph  ->  ( E. x ( ph  /\  ps )  ->  A. x
( ph  ->  ps )
) )
3 nfa1 1845 . . . . 5  |-  F/ x A. x ( ph  ->  ps )
4 ancl 546 . . . . . . 7  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ( ph  /\ 
ps ) ) )
5 simpl 457 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ph )
64, 5impbid1 203 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  <->  ( ph  /\  ps ) ) )
76sps 1814 . . . . 5  |-  ( A. x ( ph  ->  ps )  ->  ( ph  <->  (
ph  /\  ps )
) )
83, 7eubid 2296 . . . 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 228 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E! x ph  ->  E. x
( ph  /\  ps )
) )
1110com12 31 . 2  |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  ->  E. x
( ph  /\  ps )
) )
122, 11impbid 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 369   A.wal 1377   E.wex 1596   E!weu 2275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600  df-eu 2279  df-mo 2280
This theorem is referenced by: (None)
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