Theorem ax17e 1618

Description: A rephrasing of ax-17 1616 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)

Assertion

Ref	Expression
ax17e	⊢ (∃xφ → φ)

Distinct variable group:   φ,x

Proof of Theorem ax17e

Step	Hyp	Ref	Expression
1		df-ex 1542	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
2		ax-17 1616	. . 3 ⊢ (¬ φ → ∀x ¬ φ)
3	2	con1i 121	. 2 ⊢ (¬ ∀x ¬ φ → φ)
4	1, 3	sylbi 187	1 ⊢ (∃xφ → φ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-17 1616

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  exlimiv  1634  exlimdv  1636  19.9v  1664  equid  1676