Theorem imor

Description: Implication in terms of disjunction. Theorem *4.6 of WhiteheadRussell p. 120. (Contributed by NM, 3-Jan-1993)

	Ref	Expression
Assertion	imor	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( ¬ 𝜑 ∨ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		notnotb	⊢ ( 𝜑 ↔ ¬ ¬ 𝜑 )
2	1	imbi1i	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( ¬ ¬ 𝜑 → 𝜓 ) )
3		df-or	⊢ ( ( ¬ 𝜑 ∨ 𝜓 ) ↔ ( ¬ ¬ 𝜑 → 𝜓 ) )
4	2 3	bitr4i	⊢ ( ( 𝜑 → 𝜓 ) ↔ ( ¬ 𝜑 ∨ 𝜓 ) )