Mathematical Logic Proof Help -- 2

I’m working on a set of mathematical-logic assignments focused on proofs and theorems and need precise, well-structured assistance. The material ranges from classic propositional and predicate-logic proofs to more advanced topics such as completeness, soundness, and compactness.

Here’s what I’m after:
• Clear, step-by-step proofs written in rigorous mathematical style (LaTeX preferred).
• Brief explanatory notes after each major step so I can follow the reasoning.
• A short verification section confirming each argument’s validity or highlighting any hidden assumptions.

I will supply the statements, axioms, or prior lemmas; you return the finished proofs ready for submission, along with your explanatory commentary. Accuracy is critical—please ensure every deduction is logically sound and references standard theorems where appropriate. If you have a background in formal logic systems or experience tutoring similar topics, let me know in your proposal.