Theorem ax5e 1881

Description: A rephrasing of ax-5 1879 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 1879	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
2		eximal 1747	. 2 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
3	1, 2	mpbir 221	1 ⊢ (∃𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1521  ∃wex 1744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-5 1879

This theorem depends on definitions:  df-bi 197  df-ex 1745

This theorem is referenced by:  ax5ea  1882  exlimiv  1898  exlimdv  1901  19.21v  1908  19.23v  1911  19.36imv  1913  19.41v  1917  19.9v  1953  aeveq  2024  relopabi  5278  toprntopon  20777  bj-cbvexivw  32785  bj-eqs  32788  bj-snsetex  33076  bj-snglss  33083  topdifinffinlem  33325  ac6s6f  34111  fnchoice  39502