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Theorem eupickb 2635
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 2633 . . 3 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
213adant2 1130 . 2 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
3 exancom 1863 . . . 4 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
4 eupick 2633 . . . 4 ((∃!𝑥𝜓 ∧ ∃𝑥(𝜓𝜑)) → (𝜓𝜑))
53, 4sylan2b 594 . . 3 ((∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜓𝜑))
653adant1 1129 . 2 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜓𝜑))
72, 6impbid 211 1 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086  wex 1780  ∃!weu 2566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-ex 1781  df-nf 1785  df-mo 2538  df-eu 2567
This theorem is referenced by: (None)
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