Editing Propositional calculus (section)

====Proof example in P<sub>2</sub>====
As an example, a proof of <math> A \to A </math> in P<sub>2</sub> is given below. First, the axioms are given names:
:(A1) <math>(p \to (q \to p))</math>
:(A2) <math>((p \to (q \to r)) \to ((p \to q) \to (p \to r)))</math>
:(A3) <math>((\neg p \to \neg q) \to (q \to p))</math>

And the proof is as follows:
# <math> A \to ((B \to A) \to A)</math> &nbsp; &nbsp; &nbsp; (instance of (A1))
# <math> (A \to ((B \to A) \to A)) \to ((A \to (B \to A)) \to (A \to A))</math> &nbsp; &nbsp; &nbsp; (instance of (A2))
# <math> (A \to (B \to A)) \to (A \to A)</math> &nbsp; &nbsp; &nbsp; (from (1) and (2) by [[modus ponens]])
# <math> A \to (B \to A)</math> &nbsp; &nbsp; &nbsp; (instance of (A1))
# <math> A \to A </math> &nbsp; &nbsp; &nbsp; (from (4) and (3) by modus ponens)