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	$$⊢\phi \to A\in {ℝ}^{+}$$
	logne0d.1	$$⊢\phi \to A\ne 1$$
Assertion	logne0d	$$⊢\phi \to \log A\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		logne0d.a	$$⊢\phi \to A\in {ℝ}^{+}$$
2		logne0d.1	$$⊢\phi \to A\ne 1$$
3		logne0	$$⊢\left(A\in {ℝ}^{+}\wedge A\ne 1\right)\to \log A\ne 0$$
4	1 2 3	syl2anc	$$⊢\phi \to \log A\ne 0$$