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Metamath Proof Explorer

Theorem an3andi

Description: Distribution of conjunction over threefold conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019)

Ref Expression
Assertion an3andi ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜑 ∧ 𝜃 ) ) )

Proof

Step Hyp Ref Expression
1 anandi ⊢ ( ( 𝜑 ∧ ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) ∧ ( 𝜑 ∧ 𝜃 ) ) )
2 anandi ⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ) )
3 1 2 bianbi ⊢ ( ( 𝜑 ∧ ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) ) ↔ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ) ∧ ( 𝜑 ∧ 𝜃 ) ) )
4 df-3an ⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ↔ ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) )
5 4 anbi2i ⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ) ↔ ( 𝜑 ∧ ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) ) )
6 df-3an ⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜑 ∧ 𝜃 ) ) ↔ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ) ∧ ( 𝜑 ∧ 𝜃 ) ) )
7 3 5 6 3bitr4i ⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜑 ∧ 𝜃 ) ) )