Reddit reviews Mathematical Logic: A First Course (Dover Books on Mathematics)
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How about "Mathematical Logic: A first course" by Joel Robbin (https://www.amazon.com/Mathematical-Logic-First-Course-Mathematics/dp/048645018X/ref=sr_1_1?ie=UTF8&qid=1541650359&sr=8-1&keywords=joel+robbin+mathematical+logic)
He covers axiomatic symbolic logic in a system that just has F and material implication. He covers some axiomatic systems in a pretty basic way that I've not seen before.
Also, Smullyan's "Beginner's Guide to Mathematical Logic", and it's sequel, "Beginner's further guide to mathematical logic". Smullyan is obsessed with the idea of primitives representable by other primitives, and vice versa, it shows in "To Mock A Mockingbird" and these beginner's guides.