If you have a theorem $\vdash \alpha \rightarrow \beta$ and a theorem $\vdash \gamma$ then if $\alpha$ is a sub-expression of $\gamma$, and this sub-expression has an even number of negations within $\gamma$ (and it is not within a xor or boolean equality), then a new theorem can be obtained by replacing $\alpha$ with $\beta$ in $\vdash \gamma$.
Similar, if $\beta$ is a sub-expression of $\gamma$, and this has a odd number of negations, then a new theorem can be obtained by replacing $\beta$ with $\alpha$.
It is not so difficult to see that this is true, but has this rule a name? And is there any literature about it? If you introduce this rule, you can probably skip some axioms. Also, it is a more general form of modus pones.
The background of this question is, that one of my interests is how to do real mathematics in a formal logic. My opinion is that both sides should make steps to come closer to each other. Logicians should make logics that are more practical to use, and mathematicians should make proofs that are easier to formalize. I think above rule is quite natural for mathematicians, and can be verified by computer. At least when I am doing mathematics, I am well aware if a certain sub-expression is in a position that it can be weakened or strengthened. And I think that counts for all mathematicians.
With this rule you don't need the low-level proving with shifting assumptions in front of the $\vdash$ sign, to weaken or strengthen the sub-expression. This low-level shifting is a practice that in general can not be found in books of mathematics that do not deal with logic.