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Theorem ax5e 1904
Description: A rephrasing of ax-5 1902 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 1902 . 2 𝜑 → ∀𝑥 ¬ 𝜑)
2 eximal 1774 . 2 ((∃𝑥𝜑𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
31, 2mpbir 232 1 (∃𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-5 1902
This theorem depends on definitions:  df-bi 208  df-ex 1772
This theorem is referenced by:  ax5ea  1905  exlimiv  1922  exlimdv  1925  19.21v  1931  19.23v  1934  19.36imv  1937  19.41v  1941  19.9v  1979  aeveq  2052  sbv  2089  sbequ2  2241  mo4  2646  relopabi  5688  lfuhgr3  32264  bj-cbvalim  33876  bj-cbvexim  33877  bj-cbvexivw  33903  bj-eqs  33906  bj-nnfv  33981  bj-snsetex  34173  bj-snglss  34180  topdifinffinlem  34511  ac6s6f  35334  fnchoice  41166
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