Theorem abn0

Description: Nonempty class abstraction. See also ab0 . (Contributed by NM, 26-Dec-1996) (Proof shortened by Mario Carneiro, 11-Nov-2016) Avoid df-clel , ax-8 . (Revised by Gino Giotto, 30-Aug-2024)

	Ref	Expression
Assertion	abn0	⊢ ( { 𝑥 ∣ 𝜑 } ≠ ∅ ↔ ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ab0	⊢ ( { 𝑥 ∣ 𝜑 } = ∅ ↔ ∀ 𝑥 ¬ 𝜑 )
2	1	notbii	⊢ ( ¬ { 𝑥 ∣ 𝜑 } = ∅ ↔ ¬ ∀ 𝑥 ¬ 𝜑 )
3		df-ne	⊢ ( { 𝑥 ∣ 𝜑 } ≠ ∅ ↔ ¬ { 𝑥 ∣ 𝜑 } = ∅ )
4		df-ex	⊢ ( ∃ 𝑥 𝜑 ↔ ¬ ∀ 𝑥 ¬ 𝜑 )
5	2 3 4	3bitr4i	⊢ ( { 𝑥 ∣ 𝜑 } ≠ ∅ ↔ ∃ 𝑥 𝜑 )