The deductive reasoning "commonly used in math proofs on a daily base" is regimented by classical first-order logic. So that's where to start.
If you asked me to suggest just one book for beginners at logic with some mathematical background, then I think I would nowadays recommend the particularly accessible and well-written Ian Chiswell and Wilfrid Hodges, Mathematical Logic (OUP 2007) as the best starting point.
But different people with different backgrounds will like different books. As an introductory guide to just some of what is on offer, you'll find an annotated Teach Yourself Logic reading list at http://www.logicmatters.net/resources/pdfs/TeachYourselfLogic2.pdf This handout as yet only covers little more than first-order logic, but it will get you started.