Metamath Proof Explorer


Theorem 3an4anass

Description: Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018)

Ref Expression
Assertion 3an4anass ( ( ( 𝜑𝜓𝜒 ) ∧ 𝜃 ) ↔ ( ( 𝜑𝜓 ) ∧ ( 𝜒𝜃 ) ) )

Proof

Step Hyp Ref Expression
1 df-3an ( ( 𝜑𝜓𝜒 ) ↔ ( ( 𝜑𝜓 ) ∧ 𝜒 ) )
2 1 anbi1i ( ( ( 𝜑𝜓𝜒 ) ∧ 𝜃 ) ↔ ( ( ( 𝜑𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) )
3 anass ( ( ( ( 𝜑𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) ↔ ( ( 𝜑𝜓 ) ∧ ( 𝜒𝜃 ) ) )
4 2 3 bitri ( ( ( 𝜑𝜓𝜒 ) ∧ 𝜃 ) ↔ ( ( 𝜑𝜓 ) ∧ ( 𝜒𝜃 ) ) )