Theorem imdi

Description: Distributive law for implication. Compare Theorem *5.41 of WhiteheadRussell p. 125. (Contributed by NM, 5-Aug-1993)

	Ref	Expression
Assertion	imdi	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-2	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )
2		pm2.86	⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → ( 𝜓 → 𝜒 ) ) )
3	1 2	impbii	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) )