I am not sure what is a "meta axiom". Could you give an example for one?
For a moment, forget about "and" and use "zxhds" instead. You need to specify how it works.
There is nothing magic, intrinsic or natural about "truth" or "implies". It is all rules of symbol rearrangement.
As to your reply above, regardless of what you call axioms, semantically speaking there is an isomorphic subset in common for all useful axiomatic system is what I'm postulating. Can you think of an axiomatic system that doesn't have something iso to identity and equivalence for instance?
1. When it comes to the Gödel's incompleteness theorem - for non-recursive systems you may you can get without that, vide https://math.stackexchange.com/questions/3359878/what-is-som...
2. In general, you can do a lot without equivalence. For example, in set theory (ZF), you get only trivial equivalence (sets having the same elements are the same set). But you can still do a lot with "contains". A similar thing works with partially ordered sets, in which you have the "<=" operator.