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Metamath Proof Explorer

Theorem ral0

Description: Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005) Avoid df-clel , ax-8 . (Revised by GG, 2-Sep-2024)

Ref Expression
Assertion ral0 ⊢ ∀ x ∈ ∅ φ

Proof

Step Hyp Ref Expression
1 eqid ⊢ ∅ = ∅
2 rzal ⊢ ∅ = ∅ → ∀ x ∈ ∅ φ
3 1 2 ax-mp ⊢ ∀ x ∈ ∅ φ