This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 19.9v

Description: Version of 19.9 with a disjoint variable condition, requiring fewer axioms. Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v . (Contributed by NM, 28-May-1995) Remove dependency on ax-7 . (Revised by Wolf Lammen, 4-Dec-2017)

		Ref	Expression
	Assertion	19.9v	⊢ ( ∃ 𝑥 𝜑 ↔ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ax5e	⊢ ( ∃ 𝑥 𝜑 → 𝜑 )
2		19.8v	⊢ ( 𝜑 → ∃ 𝑥 𝜑 )
3	1 2	impbii	⊢ ( ∃ 𝑥 𝜑 ↔ 𝜑 )