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Theorem andir 820
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 819 . 2 ((𝜒 ∧ (𝜑𝜓)) ↔ ((𝜒𝜑) ∨ (𝜒𝜓)))
2 ancom 266 . 2 (((𝜑𝜓) ∧ 𝜒) ↔ (𝜒 ∧ (𝜑𝜓)))
3 ancom 266 . . 3 ((𝜑𝜒) ↔ (𝜒𝜑))
4 ancom 266 . . 3 ((𝜓𝜒) ↔ (𝜒𝜓))
53, 4orbi12i 765 . 2 (((𝜑𝜒) ∨ (𝜓𝜒)) ↔ ((𝜒𝜑) ∨ (𝜒𝜓)))
61, 2, 53bitr4i 212 1 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wo 709
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 710
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  anddi  822  excxor  1389  xordc1  1404  sbequilem  1849  rexun  3330  rabun2  3429  reuun2  3433  xpundir  4698  coundi  5145  mptun  5363  tpostpos  6284  ltxr  9800  pythagtriplem2  12293  pythagtrip  12310
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