Theorem idALT

Alternate proof of id. This version is proved directly from the axioms for demonstration purposes. This proof is a popular example in the literature and is identical, step for step, to the proofs of Theorem 1 of [Margaris] p. 51, Example 2.7(a) of [Hamilton] p. 31, Lemma 10.3 of [BellMachover] p. 36, and Lemma 1.8 of [Mendelson] p. 36. It is also "Our first proof" in Hirst and Hirst's A Primer for Logic and Proof p. 17 (PDF p. 23) at http://www.appstate.edu/~hirstjl/primer/hirst.pdf. Note that the second occurrences of \( \varphi \) in Steps 1 to 4 and the sixth in Step 3 may be simultaneously replaced by any wff \( \psi \), which may ease the understanding of the proof. For a shorter version of the proof that takes advantage of previously proved theorems, see id. (Contributed by NM, 30-Sep-1992.) (Proof modification is discouraged.) Use id instead. (New usage is discouraged.)

Conclusion
\( \vdash \varphi \rightarrow \varphi \)
Proof
l0
ax-1
\( \vdash \varphi \rightarrow \varphi \rightarrow \varphi \)
l1
ax-1
\( \vdash \varphi \rightarrow (\varphi \rightarrow \varphi) \rightarrow \varphi \)
ax-2
\( \vdash (\varphi \rightarrow (\varphi \rightarrow \varphi) \rightarrow \varphi) \rightarrow (\varphi \rightarrow \varphi \rightarrow \varphi) \rightarrow \varphi \rightarrow \varphi \)
ax-mp l1 _
\( \vdash (\varphi \rightarrow \varphi \rightarrow \varphi) \rightarrow \varphi \rightarrow \varphi \)
ax-mp l0 _