Theorem andi 807

Description: Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)

Assertion

Ref	Expression
andi	⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))

Proof of Theorem andi

Step	Hyp	Ref	Expression
1		orc 701	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
2		olc 700	. . 3 ⊢ ((𝜑 ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
3	1, 2	jaodan 786	. 2 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
4		orc 701	. . . 4 ⊢ (𝜓 → (𝜓 ∨ 𝜒))
5	4	anim2i 339	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ (𝜓 ∨ 𝜒)))
6		olc 700	. . . 4 ⊢ (𝜒 → (𝜓 ∨ 𝜒))
7	6	anim2i 339	. . 3 ⊢ ((𝜑 ∧ 𝜒) → (𝜑 ∧ (𝜓 ∨ 𝜒)))
8	5, 7	jaoi 705	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)) → (𝜑 ∧ (𝜓 ∨ 𝜒)))
9	3, 8	impbii 125	1 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∨ wo 697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  andir  808  anddi  810  dcim  826  dcan  918  excxor  1356  sbequilem  1810  sborv  1862  r19.43  2589  indi  3323  difindiss  3330  unrab  3347  unipr  3750  uniun  3755  unopab  4007  xpundi  4595  coundir  5041  unpreima  5545  tpostpos  6161  elni2  7122  elznn0nn  9068