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Theorem andir 774
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 773 . 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 722 . 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 670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  anddi  776  dcan  886  excxor  1324  xordc1  1339  sbequilem  1777  rexun  3203  rabun2  3302  reuun2  3306  xpundir  4534  coundi  4976  mptun  5190  tpostpos  6091  ltxr  9403
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