Theorem ax5e 1829

Description: A rephrasing of ax-5 1827 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 1827	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
2		eximal 1698	. 2 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
3	1, 2	mpbir 220	1 ⊢ (∃𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-5 1827

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by:  nfv  1830  exlimiv  1845  exlimdv  1848  19.21v  1855  19.9v  1883  aeveq  1969  aevOLD  2148  relopabi  5167  bj-ax5ea  31805  bj-cbvexivw  31847  bj-eqs  31850  bj-snsetex  32144  bj-snglss  32151  bj-toprntopon  32244  topdifinffinlem  32371  ac6s6f  33151  fnchoice  38211