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Theorem anan 35656
Description: Multiple commutations in conjunction. (Contributed by Peter Mazsa, 7-Mar-2020.)
Assertion
Ref Expression
anan ((((𝜑𝜓) ∧ 𝜒) ∧ ((𝜑𝜃) ∧ 𝜏)) ↔ ((𝜓𝜃) ∧ (𝜑 ∧ (𝜒𝜏))))

Proof of Theorem anan
StepHypRef Expression
1 an4 655 . 2 ((((𝜑𝜓) ∧ 𝜒) ∧ ((𝜑𝜃) ∧ 𝜏)) ↔ (((𝜑𝜓) ∧ (𝜑𝜃)) ∧ (𝜒𝜏)))
2 anandi 675 . . . 4 ((𝜑 ∧ (𝜓𝜃)) ↔ ((𝜑𝜓) ∧ (𝜑𝜃)))
3 ancom 464 . . . 4 ((𝜑 ∧ (𝜓𝜃)) ↔ ((𝜓𝜃) ∧ 𝜑))
42, 3bitr3i 280 . . 3 (((𝜑𝜓) ∧ (𝜑𝜃)) ↔ ((𝜓𝜃) ∧ 𝜑))
54anbi1i 626 . 2 ((((𝜑𝜓) ∧ (𝜑𝜃)) ∧ (𝜒𝜏)) ↔ (((𝜓𝜃) ∧ 𝜑) ∧ (𝜒𝜏)))
6 anass 472 . 2 ((((𝜓𝜃) ∧ 𝜑) ∧ (𝜒𝜏)) ↔ ((𝜓𝜃) ∧ (𝜑 ∧ (𝜒𝜏))))
71, 5, 63bitri 300 1 ((((𝜑𝜓) ∧ 𝜒) ∧ ((𝜑𝜃) ∧ 𝜏)) ↔ ((𝜓𝜃) ∧ (𝜑 ∧ (𝜒𝜏))))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399
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 210  df-an 400
This theorem is referenced by:  inxpxrn  35800
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