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Theorem 13an22anass 31701
Description: Associative law for four conjunctions with a triple conjunction. (Contributed by Thierry Arnoux, 21-Jan-2025.)
Assertion
Ref Expression
13an22anass ((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))

Proof of Theorem 13an22anass
StepHypRef Expression
1 an2anr 635 . . 3 (((𝜓𝜒) ∧ (𝜃𝜑)) ↔ ((𝜒𝜓) ∧ (𝜑𝜃)))
2 an2anr 635 . . . 4 (((𝜑𝜒) ∧ (𝜃𝜓)) ↔ ((𝜒𝜑) ∧ (𝜓𝜃)))
3 an4 654 . . . 4 (((𝜒𝜑) ∧ (𝜓𝜃)) ↔ ((𝜒𝜓) ∧ (𝜑𝜃)))
42, 3bitri 274 . . 3 (((𝜑𝜒) ∧ (𝜃𝜓)) ↔ ((𝜒𝜓) ∧ (𝜑𝜃)))
5 an43 656 . . 3 (((𝜑𝜒) ∧ (𝜃𝜓)) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
61, 4, 53bitr2ri 299 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜓𝜒) ∧ (𝜃𝜑)))
7 3an4anass 1105 . 2 (((𝜓𝜒𝜃) ∧ 𝜑) ↔ ((𝜓𝜒) ∧ (𝜃𝜑)))
8 ancom 461 . 2 (((𝜓𝜒𝜃) ∧ 𝜑) ↔ (𝜑 ∧ (𝜓𝜒𝜃)))
96, 7, 83bitr2ri 299 1 ((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1087
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 206  df-an 397  df-3an 1089
This theorem is referenced by:  ressply1mon1p  32652
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