Theorem abn0OLD

Description: Obsolete version of abn0 as of 30-Aug-2024. (Contributed by NM, 26-Dec-1996) (Proof shortened by Mario Carneiro, 11-Nov-2016) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	abn0OLD	⊢ ( { 𝑥 ∣ 𝜑 } ≠ ∅ ↔ ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		nfab1	⊢ Ⅎ 𝑥 { 𝑥 ∣ 𝜑 }
2	1	n0f	⊢ ( { 𝑥 ∣ 𝜑 } ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ { 𝑥 ∣ 𝜑 } )
3		abid	⊢ ( 𝑥 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜑 )
4	3	exbii	⊢ ( ∃ 𝑥 𝑥 ∈ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑥 𝜑 )
5	2 4	bitri	⊢ ( { 𝑥 ∣ 𝜑 } ≠ ∅ ↔ ∃ 𝑥 𝜑 )