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Theorem andir 743
 Description: Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
Assertion
Ref Expression
andir (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))

Proof of Theorem andir
StepHypRef Expression
1 andi 742 . 2 ((𝜒 ∧ (𝜑𝜓)) ↔ ((𝜒𝜑) ∨ (𝜒𝜓)))
2 ancom 257 . 2 (((𝜑𝜓) ∧ 𝜒) ↔ (𝜒 ∧ (𝜑𝜓)))
3 ancom 257 . . 3 ((𝜑𝜒) ↔ (𝜒𝜑))
4 ancom 257 . . 3 ((𝜓𝜒) ↔ (𝜒𝜓))
53, 4orbi12i 691 . 2 (((𝜑𝜒) ∨ (𝜓𝜒)) ↔ ((𝜒𝜑) ∨ (𝜒𝜓)))
61, 2, 53bitr4i 205 1 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   ↔ wb 102   ∨ wo 639 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640 This theorem depends on definitions:  df-bi 114 This theorem is referenced by:  anddi  745  dcan  853  excxor  1285  xordc1  1300  sbequilem  1735  rexun  3151  rabun2  3244  reuun2  3248  xpundir  4425  coundi  4850  mptun  5057  tpostpos  5910  ltxr  8796
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