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Theorem ax5e 1913
Description: A rephrasing of ax-5 1911 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 1911 . 2 𝜑 → ∀𝑥 ¬ 𝜑)
2 eximal 1784 . 2 ((∃𝑥𝜑𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
31, 2mpbir 234 1 (∃𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-5 1911
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by:  ax5ea  1914  exlimiv  1931  exlimdv  1934  19.21v  1940  19.23v  1943  19.36imv  1946  19.41v  1950  19.9v  1988  aeveq  2061  sbv  2098  sbequ2  2251  mo4  2649  relopabi  5671  lfuhgr3  32440  bj-cbvalim  34052  bj-cbvexim  34053  bj-cbvexivw  34079  bj-eqs  34082  bj-nnfv  34159  bj-snsetex  34360  bj-snglss  34367  topdifinffinlem  34725  ac6s6f  35570  fnchoice  41593
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