 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  idALT Structured version   Visualization version   GIF version

Theorem idALT 23
 Description: Alternate proof of id 22. 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 𝜑 in Steps 1 to 4 and the sixth in Step 3 may be simultaneously replaced by any wff 𝜓, which may ease the understanding of the proof. For a shorter version of the proof that takes advantage of previously proved theorems, see id 22. (Contributed by NM, 30-Sep-1992.) (Proof modification is discouraged.) Use id 22 instead. (New usage is discouraged.)
Assertion
Ref Expression
idALT (𝜑𝜑)

Proof of Theorem idALT
StepHypRef Expression
1 ax-1 6 . 2 (𝜑 → (𝜑𝜑))
2 ax-1 6 . . 3 (𝜑 → ((𝜑𝜑) → 𝜑))
3 ax-2 7 . . 3 ((𝜑 → ((𝜑𝜑) → 𝜑)) → ((𝜑 → (𝜑𝜑)) → (𝜑𝜑)))
42, 3ax-mp 5 . 2 ((𝜑 → (𝜑𝜑)) → (𝜑𝜑))
51, 4ax-mp 5 1 (𝜑𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7 This theorem is referenced by:  id1  27915
 Copyright terms: Public domain W3C validator