I am having trouble giving proofs for these theorems. I understand them intuitively but I am not sure how to explain them formally.
So for (i), if Φ, α ⊨ β then this implies that Φ ⊨ α → β because when Φ and α are true β is as well, so the formula α → β would be satisfied as both antecedent and consequent are true. If Φ ⊨ α → β then this implies that Φ,α ⊨ β because when Φ is true then so is α → β, so if Φ,α is true then α → β must still be true (since Φ is) and the only way for it to hold would be if β is also true. My problem is really in trying to formalize this into a proof by induction or some sort of more formal proof then an explanation. My best guess would be to try and prove it as a universal conditional regarding truth valuations?
(i) For any set Φ of formulas and any two formulas α and β, prove that Φ, α ⊨ β iff Φ ⊨ α → β.
(ii) Prove that {α1, α2, ... , αk} ⊨ β iff ⊨ α1 → α2 → ... → αk → β.
Thanks for the help in advance!