Theorem andir 819

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 818	. 2 ⊢ ((𝜒 ∧ (𝜑 ∨ 𝜓)) ↔ ((𝜒 ∧ 𝜑) ∨ (𝜒 ∧ 𝜓)))
2		ancom 266	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜒 ∧ (𝜑 ∨ 𝜓)))
3		ancom 266	. . 3 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜒 ∧ 𝜑))
4		ancom 266	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))
5	3, 4	orbi12i 764	. 2 ⊢ (((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜒 ∧ 𝜑) ∨ (𝜒 ∧ 𝜓)))
6	1, 2, 5	3bitr4i 212	1 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∨ wo 708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  anddi  821  dcan  933  excxor  1378  xordc1  1393  sbequilem  1838  rexun  3315  rabun2  3414  reuun2  3418  xpundir  4682  coundi  5129  mptun  5346  tpostpos  6262  ltxr  9771  pythagtriplem2  12258  pythagtrip  12275