MHB Rules of Inference for Proving p→(p→q)→(p→q)

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The discussion focuses on proving the logical statement p→(p→q)→(p→q) using a deductive system. The user has filled in most proof stages but is struggling with the last stage, specifically identifying the rule or axiom used. They have provided a table of proof stages, relevant axioms, and previously proved statements. The only inference rule available is modus ponens, and the user believes that the last stage follows from applying modus ponens to earlier stages. Assistance is sought to confirm this reasoning and complete the proof.
Yankel
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Hello to you all,

I am trying to prove the following:

\[\vdash \left ( p\rightarrow \left ( p\rightarrow q \right ) \right )\rightarrow \left ( p\rightarrow q \right )\]

I was given a table with the proof stages, and I had to fill the blanks. Sometimes the blanks were the rule or axiom used in this stage, and sometimes it was the result of using the rule/axiom. I filled all, but I can't figure out the last stage. More specifically, I can't figure out which rule / axiom was used in the last stage (stage 7). I have completed all other stages, and I think it's correct.

I am attaching as figures, the table of the proof, the 3 axioms I am allowed to use (I am using the L deductive system), and 4 statements which were proved already and can be used. In addition, the only inference rule is the modus ponens.

Thank you in advance for helping me complete the last stage.

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I think 7 follows by MP from 3 and 6.
 
I think you are correct, I didn't see it. Thank you ! (Yes)
 
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