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Theorem anandir 677
Description: Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
anandir (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))

Proof of Theorem anandir
StepHypRef Expression
1 anidm 564 . . 3 ((𝜒𝜒) ↔ 𝜒)
21anbi2i 623 . 2 (((𝜑𝜓) ∧ (𝜒𝜒)) ↔ ((𝜑𝜓) ∧ 𝜒))
3 an4 656 . 2 (((𝜑𝜓) ∧ (𝜒𝜒)) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
42, 3bitr3i 277 1 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 207  df-an 396
This theorem is referenced by:  anandi3r  1102  disjxun  5146  fununi  6643  imadif  6652  elfzuzb  13555  frgr3v  30304  5oalem3  31685  5oalem5  31687  refrelredund4  38617  nzin  44314  un2122  44788
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