Theorem ax5e 1871

Description: A rephrasing of ax-5 1869 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 1869	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
2		eximal 1745	. 2 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
3	1, 2	mpbir 223	1 ⊢ (∃𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1505  ∃wex 1742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-5 1869

This theorem depends on definitions:  df-bi 199  df-ex 1743

This theorem is referenced by:  ax5ea  1872  exlimiv  1889  exlimdv  1892  19.21v  1898  19.23v  1901  19.36imv  1904  19.41v  1908  19.9v  1938  aeveq  2005  sbv  2040  relopabi  5544  bj-cbvalim  33492  bj-cbvexim  33493  bj-cbvexivw  33521  bj-eqs  33524  bj-nnfv  33786  bj-snsetex  33799  bj-snglss  33806  topdifinffinlem  34076  ac6s6f  34901  fnchoice  40711