Theorem rabn0

Description: Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999) (Revised by BJ, 16-Jul-2021)

	Ref	Expression
Assertion	rabn0	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ≠ ∅ ↔ ∃ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		rabeq0	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } = ∅ ↔ ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 )
2	1	necon3abii	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ≠ ∅ ↔ ¬ ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 )
3		dfrex2	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 )
4	2 3	bitr4i	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ≠ ∅ ↔ ∃ 𝑥 ∈ 𝐴 𝜑 )