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Theorem ax5e 2007
Description: A rephrasing of ax-5 2005 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 2005 . 2 𝜑 → ∀𝑥 ¬ 𝜑)
2 eximal 1877 . 2 ((∃𝑥𝜑𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
31, 2mpbir 222 1 (∃𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1650  wex 1874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-5 2005
This theorem depends on definitions:  df-bi 198  df-ex 1875
This theorem is referenced by:  ax5ea  2008  exlimiv  2025  exlimdv  2028  19.21v  2034  19.23v  2037  19.36imv  2040  19.41v  2044  19.9v  2078  aeveq  2149  relopabi  5414  toprntopon  21009  bj-cbvexivw  33096  bj-eqs  33099  bj-snsetex  33378  bj-snglss  33385  topdifinffinlem  33628  wl-dv-sb  33755  ac6s6f  34402  fnchoice  39840
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