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Theorem eupickb 2348
Description: Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
eupickb  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph 
<->  ps ) )

Proof of Theorem eupickb
StepHypRef Expression
1 eupick 2346 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
213adant2 977 . 2  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
3 3simpc 957 . . 3  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( E! x ps  /\  E. x ( ph  /\  ps ) ) )
4 pm3.22 438 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ps  /\  ph ) )
54eximi 1586 . . . 4  |-  ( E. x ( ph  /\  ps )  ->  E. x
( ps  /\  ph ) )
65anim2i 554 . . 3  |-  ( ( E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( E! x ps  /\  E. x ( ps  /\  ph ) ) )
7 eupick 2346 . . 3  |-  ( ( E! x ps  /\  E. x ( ps  /\  ph ) )  ->  ( ps  ->  ph ) )
83, 6, 73syl 19 . 2  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ps  ->  ph ) )
92, 8impbid 185 1  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph 
<->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551   E!weu 2283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288
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