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Theorem eupickb 2029
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 2027 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
213adant2 962 . 2  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
3 3simpc 942 . . 3  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( E! x ps  /\  E. x ( ph  /\  ps ) ) )
4 pm3.22 261 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ps  /\  ph ) )
54eximi 1536 . . . 4  |-  ( E. x ( ph  /\  ps )  ->  E. x
( ps  /\  ph ) )
65anim2i 334 . . 3  |-  ( ( E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( E! x ps  /\  E. x ( ps  /\  ph ) ) )
7 eupick 2027 . . 3  |-  ( ( E! x ps  /\  E. x ( ps  /\  ph ) )  ->  ( ps  ->  ph ) )
83, 6, 73syl 17 . 2  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ps  ->  ph ) )
92, 8impbid 127 1  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph 
<->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924   E.wex 1426   E!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by: (None)
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