Prove Theorems 2.50-2.54 on logical consequence. Theorem 2.50 UA if and only if A; Ai A. Definition 2.511 A; is an abbreviation for A ^ .. ^ An. The notation ; is used if the bounds are obvious from the context or if the set of formulas is infinite. A similar notation Vis used for disjunction. Example 2.52 From Example 2.49, {p,q} = (pvr) ^ (q vr), so by Theo- rem 2.50,= (p ^q) (pvr) ^ (q vr). The proof of Theorem 2.50, as well as the proofs of the following two theorems are left as exercises. Theorem 2.53 If U = A then UU{B} = A for any formula B. Theorem 2.54 If U = A and B is valid then U {B} | A.