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	⊢ (∃𝑥𝜑 → 𝜑)

Distinct variable group:   𝜑,𝑥

Proof of Theorem ax5e

Step	Hyp	Ref	Expression
1		ax-5 1911	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
2		eximal 1783	. 2 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
3	1, 2	mpbir 233	1 ⊢ (∃𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  ∃wex 1780

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 209  df-ex 1781

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  2250  mo4  2650  relopabi  5696  lfuhgr3  32368  bj-cbvalim  33980  bj-cbvexim  33981  bj-cbvexivw  34007  bj-eqs  34010  bj-nnfv  34085  bj-snsetex  34277  bj-snglss  34284  topdifinffinlem  34630  ac6s6f  35453  fnchoice  41293