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Given Lukasiewicz axiom system for Classical Propositional Logic (CPL):

(L1) α→(β→α)

(L2) (α→(β→γ))→(α→β)→(α→γ)

(L3) (¬α→¬β)→(β→α)

and the usual Modus Ponens, does my proof of (P→Q)→((Q→R)→(P→R)) below correct? Can someone point out where I made mistake?

(1) (P→R)→((Q→R)→(P→R)) (L1)

(2) (P→(Q→R))→((P→Q)→(P→R)) (L2)

(3) (Q→R)→(P→(Q→R)) (L1)

(4) (P→Q)→(P→R) (2,3 MP)

(5) (Q→R)→(P→R) (1,4 MP)

(6) (P→Q)→((Q→R)→(P→R)) (4,5 MP)

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    $\begingroup$ How does your step (4) work? The only deduction I can see from (2), (3) is the derived rule HS giving $(Q\to R)\to((P\to Q)\to(P\to R))$. $\endgroup$ – user10354138 May 22 at 5:27
  • $\begingroup$ You can find the proof in the answer to this post : T2. $\endgroup$ – Mauro ALLEGRANZA May 22 at 6:15
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No. Steps 4,5, and 6 are all not correct applications of MP

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