Prove [(p=>q) ^ (q=>r)] => (p=>r) is a Tautology
Proof: " "
1 = [(~pvq) ^ (~qvr)] => (~pvr) by OULE p=>q = ~pvr
2 =~[(~pvq) ^ (~qvr)} v (~pvr) by OULE " "
3 =[~(~pvq) v ~(~qvr)] v (~pvr) by De Morgan's Law
4 =[(~~p^~q) v (~~q^~r)] v (~pvr) by De Morgan's Law
5 =[(p^~q) v (q^~r)] v (~pvr) by Double Negation Law
6 =[((p^~q) v q ^ ((p^~q) v ~r)] v (~pvr) by Distributive Law
7 =[((pvq) ^ (~qvq)) ^ ((p^~q) v ~r))] v (~pvr) by Distributive Law
8 =[((pvq) ^ T) ^ ((p^~q) v ~r)] v (~pvr) by Negation law
9 =[(pvq) ^ ((p^~q) v ~r)] v (~pvr) by Identity law
10 =(~pvr) v [(pvq) ^ ((p^~q) v ~r)] by Commutative Law
11 =[(~pvr) v (pvq)[ ^ [(~pvr) v ((p^~q) v ~r] by Distributive Law
12 =[(~pvp) v (rvq)] ^ [(rv~r) v (~pv(p^~q))] by Commutative Law
13 =[T v (rvq)] ^ [Tv(~pv(p^~q))] by Negation Law
14 =T^T by Domination Law
15 =T by Idempotent Law or Definition of Conjunction
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