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Theorem eupicka 2525
Description: Version of eupick 2524 with closed formulas. (Contributed by NM, 6-Sep-2008.)
Assertion
Ref Expression
eupicka ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))

Proof of Theorem eupicka
StepHypRef Expression
1 nfeu1 2468 . . 3 𝑥∃!𝑥𝜑
2 nfe1 2014 . . 3 𝑥𝑥(𝜑𝜓)
31, 2nfan 1816 . 2 𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓))
4 eupick 2524 . 2 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
53, 4alrimi 2069 1 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1473  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-11 2021  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463
This theorem is referenced by:  eupickbi  2527  frege124d  36866  sbiota1  37451
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