The mutual co-implication of thermodynamics' first and second laws

In classical phenomenological thermodynamics the first and second laws can be regarded as independent statements. Statistical mechanics provides a microscopic substratum that explains thermodynamics in probabilistic terms via a microstate probability distribution ${p_i}$. We study here a hitherto unexplored microscopic connection between the two laws. Given an information measure (or entropic form), each of the two laws implies the other through the process $p_i \to p_i+dp_i$.