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  2560  rspn0  4322  relopabi  5788  lfuhgr3  35114  bj-cbvalim  36640  bj-cbvexim  36641  bj-cbvexivw  36667  bj-eqs  36670  bj-nnfv  36749  bj-snsetex  36958  bj-snglss  36965  topdifinffinlem  37342  ac6s6f  38174  ismnushort  44297  fnchoice  45030  ormklocald  46879  natlocalincr  46881