Theorem andir 809

Description: Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)

Assertion

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

Proof of Theorem andir

Step	Hyp	Ref	Expression
1		andi 808	. 2 ⊢ ((𝜒 ∧ (𝜑 ∨ 𝜓)) ↔ ((𝜒 ∧ 𝜑) ∨ (𝜒 ∧ 𝜓)))
2		ancom 264	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜒 ∧ (𝜑 ∨ 𝜓)))
3		ancom 264	. . 3 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜒 ∧ 𝜑))
4		ancom 264	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))
5	3, 4	orbi12i 754	. 2 ⊢ (((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜒 ∧ 𝜑) ∨ (𝜒 ∧ 𝜓)))
6	1, 2, 5	3bitr4i 211	1 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∨ wo 698

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 699

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  anddi  811  dcan  923  excxor  1368  xordc1  1383  sbequilem  1826  rexun  3301  rabun2  3400  reuun2  3404  xpundir  4660  coundi  5104  mptun  5318  tpostpos  6228  ltxr  9707  pythagtriplem2  12194  pythagtrip  12211