Theorem imdistan 571

Description: Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)

Assertion

Ref	Expression
imdistan	⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒)))

Proof of Theorem imdistan

Step	Hyp	Ref	Expression
1		pm5.42 547	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒))))
2		impexp 454	. 2 ⊢ (((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒)) ↔ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒))))
3	1, 2	bitr4i 281	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 400

This theorem is referenced by:  imdistand  574  rmoim  3679  ss2rab  3998  marypha2lem3  8885  inxpss3  35731