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Theorem eupickb 2360
Description: Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
Assertion
Ref Expression
eupickb  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph 
<->  ps ) )

Proof of Theorem eupickb
StepHypRef Expression
1 eupick 2358 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
213adant2 1015 . 2  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
3 exancom 1672 . . . 4  |-  ( E. x ( ph  /\  ps )  <->  E. x ( ps 
/\  ph ) )
4 eupick 2358 . . . 4  |-  ( ( E! x ps  /\  E. x ( ps  /\  ph ) )  ->  ( ps  ->  ph ) )
53, 4sylan2b 475 . . 3  |-  ( ( E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ps  ->  ph ) )
653adant1 1014 . 2  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ps  ->  ph ) )
72, 6impbid 191 1  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph 
<->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   E.wex 1613   E!weu 2283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-12 1855  ax-13 2000
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-ex 1614  df-nf 1618  df-eu 2287  df-mo 2288
This theorem is referenced by: (None)
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