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Theorem andir 766
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 765 . 2 ((𝜒 ∧ (𝜑𝜓)) ↔ ((𝜒𝜑) ∨ (𝜒𝜓)))
2 ancom 262 . 2 (((𝜑𝜓) ∧ 𝜒) ↔ (𝜒 ∧ (𝜑𝜓)))
3 ancom 262 . . 3 ((𝜑𝜒) ↔ (𝜒𝜑))
4 ancom 262 . . 3 ((𝜓𝜒) ↔ (𝜒𝜓))
53, 4orbi12i 714 . 2 (((𝜑𝜒) ∨ (𝜓𝜒)) ↔ ((𝜒𝜑) ∨ (𝜒𝜓)))
61, 2, 53bitr4i 210 1 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wo 662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  anddi  768  dcan  878  excxor  1312  xordc1  1327  sbequilem  1763  rexun  3169  rabun2  3267  reuun2  3271  xpundir  4462  coundi  4895  mptun  5106  tpostpos  5977  ltxr  9171
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