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Theorem eupicka 2625
Description: Version of eupick 2624 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 2577 . . 3 𝑥∃!𝑥𝜑
2 nfe1 2139 . . 3 𝑥𝑥(𝜑𝜓)
31, 2nfan 1894 . 2 𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓))
4 eupick 2624 . 2 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
53, 4alrimi 2201 1 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1531  wex 1773  ∃!weu 2557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2166
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-mo 2529  df-eu 2558
This theorem is referenced by:  eupickbi  2627  frege124d  43222  sbiota1  43902
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