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Theorem andir 808
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 807 . 2 ((𝜒 ∧ (𝜑𝜓)) ↔ ((𝜒𝜑) ∨ (𝜒𝜓)))
2 ancom 264 . 2 (((𝜑𝜓) ∧ 𝜒) ↔ (𝜒 ∧ (𝜑𝜓)))
3 ancom 264 . . 3 ((𝜑𝜒) ↔ (𝜒𝜑))
4 ancom 264 . . 3 ((𝜓𝜒) ↔ (𝜒𝜓))
53, 4orbi12i 753 . 2 (((𝜑𝜒) ∨ (𝜓𝜒)) ↔ ((𝜒𝜑) ∨ (𝜒𝜓)))
61, 2, 53bitr4i 211 1 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wo 697
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 698
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  anddi  810  dcan  918  excxor  1356  xordc1  1371  sbequilem  1810  rexun  3256  rabun2  3355  reuun2  3359  xpundir  4596  coundi  5040  mptun  5254  tpostpos  6161  ltxr  9574
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