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Theorem an3andi 1607
Description: Distribution of conjunction over threefold conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)
Assertion
Ref Expression
an3andi ((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜃)))

Proof of Theorem an3andi
StepHypRef Expression
1 df-3an 1110 . . . 4 ((𝜓𝜒𝜃) ↔ ((𝜓𝜒) ∧ 𝜃))
21anbi2i 617 . . 3 ((𝜑 ∧ (𝜓𝜒𝜃)) ↔ (𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)))
3 anandi 667 . . 3 ((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) ↔ ((𝜑 ∧ (𝜓𝜒)) ∧ (𝜑𝜃)))
4 anandi 667 . . . 4 ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
54anbi1i 618 . . 3 (((𝜑 ∧ (𝜓𝜒)) ∧ (𝜑𝜃)) ↔ (((𝜑𝜓) ∧ (𝜑𝜒)) ∧ (𝜑𝜃)))
62, 3, 53bitri 289 . 2 ((𝜑 ∧ (𝜓𝜒𝜃)) ↔ (((𝜑𝜓) ∧ (𝜑𝜒)) ∧ (𝜑𝜃)))
7 df-3an 1110 . 2 (((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜃)) ↔ (((𝜑𝜓) ∧ (𝜑𝜒)) ∧ (𝜑𝜃)))
86, 7bitr4i 270 1 ((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385  w3a 1108
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 199  df-an 386  df-3an 1110
This theorem is referenced by:  raltpd  4502
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