Theorem imdi 394

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		pm2.86 110	. 2 ⊢ (((𝜑 → 𝜓) → (𝜑 → 𝜒)) → (𝜑 → (𝜓 → 𝜒)))
3	1, 2	impbii 212	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ ((𝜑 → 𝜓) → (𝜑 → 𝜒)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  pm5.41  395  orimdi  944  bnj1174  35415  ifpim23g  44254