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Theorem anandir 533
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 382 . . 3 ((𝜒𝜒) ↔ 𝜒)
21anbi2i 438 . 2 (((𝜑𝜓) ∧ (𝜒𝜒)) ↔ ((𝜑𝜓) ∧ 𝜒))
3 an4 528 . 2 (((𝜑𝜓) ∧ (𝜒𝜒)) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
42, 3bitr3i 179 1 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  anandi3r  910  fununi  4995  imadiflem  5006  imadif  5007  imainlem  5008  elfzuzb  8986
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