Theorem ax5e 1912

Description: A rephrasing of ax-5 1910 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 1910	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
2		eximal 1782	. 2 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
3	1, 2	mpbir 231	1 ⊢ (∃𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-5 1910

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by:  ax5ea  1913  exlimiv  1930  exlimdv  1933  19.21v  1939  19.23v  1942  19.36imv  1945  19.41v  1949  19.9v  1984  aeveq  2057  sbv  2089  sbequ2  2250  mo4  2559  rspn0  4309  relopabi  5769  lfuhgr3  35092  bj-cbvalim  36618  bj-cbvexim  36619  bj-cbvexivw  36645  bj-eqs  36648  bj-nnfv  36727  bj-snsetex  36936  bj-snglss  36943  topdifinffinlem  37320  ac6s6f  38152  ismnushort  44274  fnchoice  45007  ormklocald  46856  natlocalincr  46858