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Theorem eupickb 1981
Description: Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
eupickb ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))

Proof of Theorem eupickb
StepHypRef Expression
1 eupick 1979 . . 3 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
213adant2 923 . 2 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
3 3simpc 903 . . 3 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)))
4 pm3.22 252 . . . . 5 ((𝜑𝜓) → (𝜓𝜑))
54eximi 1491 . . . 4 (∃𝑥(𝜑𝜓) → ∃𝑥(𝜓𝜑))
65anim2i 324 . . 3 ((∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (∃!𝑥𝜓 ∧ ∃𝑥(𝜓𝜑)))
7 eupick 1979 . . 3 ((∃!𝑥𝜓 ∧ ∃𝑥(𝜓𝜑)) → (𝜓𝜑))
83, 6, 73syl 17 . 2 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜓𝜑))
92, 8impbid 120 1 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885  wex 1381  ∃!weu 1900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904
This theorem is referenced by: (None)
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