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Theorem eupicka 2634
Description: Version of eupick 2633 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 2586 . . 3 𝑥∃!𝑥𝜑
2 nfe1 2146 . . 3 𝑥𝑥(𝜑𝜓)
31, 2nfan 1901 . 2 𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓))
4 eupick 2633 . 2 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
53, 4alrimi 2205 1 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1538  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-11 2153  ax-12 2170
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-mo 2538  df-eu 2567
This theorem is referenced by:  eupickbi  2636  frege124d  41690  sbiota1  42373
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