Theorem andi 813

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 707	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
2		olc 706	. . 3 ⊢ ((𝜑 ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
3	1, 2	jaodan 792	. 2 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒)))
4		orc 707	. . . 4 ⊢ (𝜓 → (𝜓 ∨ 𝜒))
5	4	anim2i 340	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ (𝜓 ∨ 𝜒)))
6		olc 706	. . . 4 ⊢ (𝜒 → (𝜓 ∨ 𝜒))
7	6	anim2i 340	. . 3 ⊢ ((𝜑 ∧ 𝜒) → (𝜑 ∧ (𝜓 ∨ 𝜒)))
8	5, 7	jaoi 711	. 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 703

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 704

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  andir  814  anddi  816  dcim  836  dcan  928  excxor  1373  sbequilem  1831  sborv  1883  r19.43  2628  indi  3374  difindiss  3381  unrab  3398  unipr  3810  uniun  3815  unopab  4068  xpundi  4667  coundir  5113  unpreima  5621  tpostpos  6243  elni2  7276  elznn0nn  9226