Theorem ax5e 1841

Description: A rephrasing of ax-5 1839 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 1839	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
2		eximal 1707	. 2 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
3	1, 2	mpbir 221	1 ⊢ (∃𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  ax5ea  1842  exlimiv  1858  exlimdv  1861  19.21v  1868  19.9v  1896  aeveq  1982  aevOLD  2162  relopabi  5245  toprntopon  20729  bj-cbvexivw  32660  bj-eqs  32663  bj-snsetex  32951  bj-snglss  32958  topdifinffinlem  33195  ac6s6f  33981  fnchoice  39188