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Theorem eupickbi 2630
Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
Assertion
Ref Expression
eupickbi (∃!𝑥𝜑 → (∃𝑥(𝜑𝜓) ↔ ∀𝑥(𝜑𝜓)))

Proof of Theorem eupickbi
StepHypRef Expression
1 eupicka 2628 . . 3 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))
21ex 411 . 2 (∃!𝑥𝜑 → (∃𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓)))
3 euex 2569 . . 3 (∃!𝑥𝜑 → ∃𝑥𝜑)
4 exintr 1893 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
53, 4syl5com 31 . 2 (∃!𝑥𝜑 → (∀𝑥(𝜑𝜓) → ∃𝑥(𝜑𝜓)))
62, 5impbid 211 1 (∃!𝑥𝜑 → (∃𝑥(𝜑𝜓) ↔ ∀𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1537  wex 1779  ∃!weu 2560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-11 2152  ax-12 2169
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-mo 2532  df-eu 2561
This theorem is referenced by:  mopickr  37535  sbaniota  43496
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