Theorem andir 1001

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 1000	. 2 ⊢ ((𝜒 ∧ (𝜑 ∨ 𝜓)) ↔ ((𝜒 ∧ 𝜑) ∨ (𝜒 ∧ 𝜓)))
2		ancom 461	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜒 ∧ (𝜑 ∨ 𝜓)))
3		ancom 461	. . 3 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜒 ∧ 𝜑))
4		ancom 461	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))
5	3, 4	orbi12i 907	. 2 ⊢ (((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜒 ∧ 𝜑) ∨ (𝜒 ∧ 𝜓)))
6	1, 2, 5	3bitr4i 304	1 ⊢ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∨ wo 842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843

This theorem is referenced by:  anddi  1003  cases  1033  cador  1588  rexun  4082  rabun2  4197  reuun2  4201  xpundir  5499  coundi  5967  mptun  6354  tpostpos  7754  wemapsolem  8850  ltxr  12349  hashbclem  13646  hashf1lem2  13650  pythagtriplem2  15971  pythagtrip  15988  vdwapun  16127  legtrid  26047  colinearalg  26367  vtxdun  26934  elimifd  29966  satfvsuclem2  32171  satf0  32181  dfon2lem5  32585  seglelin  33131  poimirlem30  34399  poimirlem31  34400  cnambfre  34417  expdioph  39056  rp-isfinite6  39320  uneqsn  39809