Theorem sepgi

Description: Inference associated with sepg . The requirement that y not occur in ph is necessary, as notsep shows. (Contributed by NM, 21-Jun-1993) (Revised by BJ, 14-Jul-2026)

	Ref	Expression
Hypothesis	sepgi.1	⊢ 𝐴 ∈ V
Assertion	sepgi	⊢ ∃ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sepgi.1	⊢ 𝐴 ∈ V
2		sepg	⊢ ( 𝐴 ∈ V → ∃ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
3	1 2	ax-mp	⊢ ∃ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )