Theorem imdistan 444

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		pm5.42 320	. 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒))))
2		impexp 263	. 2 ⊢ (((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒)) ↔ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒))))
3	1, 2	bitr4i 187	1 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  imdistand  447  pm5.3  475  rmoim  2940  ss2rab  3233  bezoutlembi  12009