This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )