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Theorem eupickbi 2661
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 2659 . . 3 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))
21ex 401 . 2 (∃!𝑥𝜑 → (∃𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓)))
3 euex 2591 . . 3 (∃!𝑥𝜑 → ∃𝑥𝜑)
4 exintr 1990 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
53, 4syl5com 31 . 2 (∃!𝑥𝜑 → (∀𝑥(𝜑𝜓) → ∃𝑥(𝜑𝜓)))
62, 5impbid 203 1 (∃!𝑥𝜑 → (∃𝑥(𝜑𝜓) ↔ ∀𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650  wex 1874  ∃!weu 2581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-mo 2565  df-eu 2582
This theorem is referenced by:  sbaniota  39312
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