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Theorem eupickb 2526
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 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))

Proof of Theorem eupickb
StepHypRef Expression
1 eupick 2524 . . 3 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
213adant2 1073 . 2 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
3 exancom 1774 . . . 4 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
4 eupick 2524 . . . 4 ((∃!𝑥𝜓 ∧ ∃𝑥(𝜓𝜑)) → (𝜓𝜑))
53, 4sylan2b 491 . . 3 ((∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜓𝜑))
653adant1 1072 . 2 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜓𝜑))
72, 6impbid 201 1 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031  wex 1695  ∃!weu 2458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463
This theorem is referenced by: (None)
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