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Theorem anandir 673
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 565 . . 3 ((𝜒𝜒) ↔ 𝜒)
21anbi2i 622 . 2 (((𝜑𝜓) ∧ (𝜒𝜒)) ↔ ((𝜑𝜓) ∧ 𝜒))
3 an4 652 . 2 (((𝜑𝜓) ∧ (𝜒𝜒)) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
42, 3bitr3i 278 1 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396
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 208  df-an 397
This theorem is referenced by:  anandi3r  1097  disjxun  5060  fununi  6425  imadif  6434  elfzuzb  12895  frgr3v  27971  5oalem3  29350  5oalem5  29352  refrelredund4  35740  nzin  40518  un2122  40992
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