I've to show via a calculation that ((P => (Q v R)) ^ (Q => R)) => (P => R) is a tautology. Now, to show this, I've to show ((P => (Q v R)) ^ (Q => R)) |= (P =>R).

I've tried the following:

(P => (Q v R)) ^ (Q => R)

|= { by ^ v weakening }

P => (Q v R)

= { by distribution }

(P => Q) v (P => R)

But now I'm stuck, since ((P => Q) v (P => R)) is actually a weaker proposition than (P => R).

Obviously I'm doing something wrong. Should I apply monotonicity? Any help would be greatly appreciated!