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Theorem ax5e 1881
 Description: A rephrasing of ax-5 1879 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
Assertion
Ref Expression
ax5e (∃𝑥𝜑𝜑)
Distinct variable group:   𝜑,𝑥

Proof of Theorem ax5e
StepHypRef Expression
1 ax-5 1879 . 2 𝜑 → ∀𝑥 ¬ 𝜑)
2 eximal 1747 . 2 ((∃𝑥𝜑𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
31, 2mpbir 221 1 (∃𝑥𝜑𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1521  ∃wex 1744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-5 1879 This theorem depends on definitions:  df-bi 197  df-ex 1745 This theorem is referenced by:  ax5ea  1882  exlimiv  1898  exlimdv  1901  19.21v  1908  19.23v  1911  19.36imv  1913  19.41v  1917  19.9v  1953  aeveq  2024  relopabi  5278  toprntopon  20777  bj-cbvexivw  32785  bj-eqs  32788  bj-snsetex  33076  bj-snglss  33083  topdifinffinlem  33325  ac6s6f  34111  fnchoice  39502
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