Theorem n0limd

Description: Deduction rule for nonempty classes. (Contributed by Thierry Arnoux, 3-Aug-2025)

	Ref	Expression
Hypotheses	n0limd.1	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
	n0limd.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝜓 )
Assertion	n0limd	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		n0limd.1	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
2		n0limd.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝜓 )
3		n0	⊢ ( 𝐴 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 )
4	1 3	sylib	⊢ ( 𝜑 → ∃ 𝑥 𝑥 ∈ 𝐴 )
5	4 2	exlimddv	⊢ ( 𝜑 → 𝜓 )