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Metamath Proof Explorer

Theorem logne0d

Description: Deduction form of logne0 . See logccne0d for a more general version. (Contributed by SN, 25-Apr-2025)

Ref Expression
Hypotheses logne0d.a ⊢ φ → A ∈ ℝ +
logne0d.1 ⊢ φ → A ≠ 1
Assertion logne0d ⊢ φ → log ⁡ A ≠ 0

Proof

Step Hyp Ref Expression
1 logne0d.a ⊢ φ → A ∈ ℝ +
2 logne0d.1 ⊢ φ → A ≠ 1
3 logne0 ⊢ A ∈ ℝ + ∧ A ≠ 1 → log ⁡ A ≠ 0
4 1 2 3 syl2anc ⊢ φ → log ⁡ A ≠ 0