In Schoenfeld's (1976) Paper:
"Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II",
it is shown in Corollary 1. (6.18) that if the Riemann Hypothesis holds, then :
$$|\pi (x) - \text{Li}(x)| < \frac{\sqrt x \ln(x)}{8\pi},~~~~~\text{for}~~~~~x \geq 2657$$
Right under it, (6.19) :
$$\pi (x) - \text{Li}(x) < \frac{\sqrt x\ln(x)}{8\pi},~~~~~\text{for}~~~~~ x\geq \frac{3}{2}$$
My question is, are these both true if and only if the Riemann Hypothesis is true, or does the phrase "If the Riemann Hypothesis holds, then" only apply for the first (6.18) statement? That is, if either one of them is proven or debunked, then does it follow that the Riemann Hypothesis is also proven or debunked?
The phrase "If the Riemann hypothesis is true, then" applies as an assumption for both (6.18) and (6.19). But: Note that you only have "if", not "only if", that is, we have, if ($\sf RH$) denotes Riemanns' hypothesis, that $$ ({\sf RH}) \implies (6.18), \quad ({\sf RH}) \implies (6.19) $$ which means that if $(\sf RH)$ is proven, both (6.18) and (6.19) are, and, by contraposition $$ \neg (6.18) \implies \neg ({\sf RH}), \quad \neg (6.19) \implies \neg ({\sf RH}) $$ That is, if (6.18) or (6.19) is disproven, also ($\sf RH$) is. Note that proving (6.18) or (6.19) does not imply anything about the truth of $(\sf RH)$.