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Theorem eupickb 2023
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 2021 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
213adant2 958 . 2  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
3 3simpc 938 . . 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 1532 . . . 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 2021 . . 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 920   E.wex 1422   E!weu 1942
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946
This theorem is referenced by: (None)
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