The difference between implication vs consequence.

Question

Here in this example a) I want to know the difference between implication($\Rightarrow$) and consequence ($\vdash$)

Does this exersice want me to show that part 2 is the consequence of part 1?

Answer 1

Implication is just a binary logical operation. Consequence means deduction of a formula from a set of formulae using axioms, theorems already proved, and logical rules like modus ponens. Moreover, logical operations have a higher binding (priority) than deduction; this makes parenthesis superfluous and the whole deduction better readable.