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Theorem anandir 907
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 679 . . 3 ((𝜒𝜒) ↔ 𝜒)
21anbi2i 732 . 2 (((𝜑𝜓) ∧ (𝜒𝜒)) ↔ ((𝜑𝜓) ∧ 𝜒))
3 an4 900 . 2 (((𝜑𝜓) ∧ (𝜒𝜒)) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
42, 3bitr3i 266 1 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383
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 197  df-an 385
This theorem is referenced by:  anandi3r  1093  disjxun  4794  fununi  6117  imadif  6126  wfrlem5  7580  elfzuzb  12521  frgr3v  27421  5oalem3  28816  5oalem5  28818  frrlem5  32082  nzin  39011  un2122  39511
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