Theorem id

Description: Principle of identity. Theorem *2.08 of WhiteheadRussell p. 101. For another version of the proof directly from axioms, see idALT . Its associated inference, idi , requires no axioms for its proof, contrary to id . Note that the second occurrences of ph in Steps 1 and 2 may be simultaneously replaced by any wff ps , which may ease the understanding of the proof. (Contributed by NM, 29-Dec-1992) (Proof shortened by Stefan Allan, 20-Mar-2006)

	Ref	Expression
Assertion	id	⊢ ( 𝜑 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ax-1	⊢ ( 𝜑 → ( 𝜑 → 𝜑 ) )
2		ax-1	⊢ ( 𝜑 → ( ( 𝜑 → 𝜑 ) → 𝜑 ) )
3	1 2	mpd	⊢ ( 𝜑 → 𝜑 )