Theorem imdi 249

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

Assertion

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

Proof of Theorem imdi

Step	Hyp	Ref	Expression
1		ax-2 7	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
2		pm2.86 100	. 2 ⊢ (((𝜑 → 𝜓) → (𝜑 → 𝜒)) → (𝜑 → (𝜓 → 𝜒)))
3	1, 2	impbii 125	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ ((𝜑 → 𝜓) → (𝜑 → 𝜒)))

Syntax hints:   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm5.41  250  tfi  4559