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Metamath Proof Explorer

Theorem bnj345

Description: /\ -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (Proof shortened by Andrew Salmon, 14-Jun-2011) (New usage is discouraged.)

Ref Expression
Assertion bnj345 ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃 ) ↔ ( 𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒 ) )

Proof

Step Hyp Ref Expression
1 bnj334 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃 ) ↔ ( 𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃 ) )
2 bnj250 ⊢ ( ( 𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃 ) ↔ ( 𝜒 ∧ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜃 ) ) )
3 3anass ⊢ ( ( 𝜒 ∧ ( 𝜑 ∧ 𝜓 ) ∧ 𝜃 ) ↔ ( 𝜒 ∧ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜃 ) ) )
4 2 3 bitr4i ⊢ ( ( 𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃 ) ↔ ( 𝜒 ∧ ( 𝜑 ∧ 𝜓 ) ∧ 𝜃 ) )
5 3anrev ⊢ ( ( 𝜒 ∧ ( 𝜑 ∧ 𝜓 ) ∧ 𝜃 ) ↔ ( 𝜃 ∧ ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
6 bnj250 ⊢ ( ( 𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( 𝜃 ∧ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ) )
7 3anass ⊢ ( ( 𝜃 ∧ ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( 𝜃 ∧ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ) )
8 6 7 bitr4i ⊢ ( ( 𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( 𝜃 ∧ ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
9 5 8 bitr4i ⊢ ( ( 𝜒 ∧ ( 𝜑 ∧ 𝜓 ) ∧ 𝜃 ) ↔ ( 𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒 ) )
10 1 4 9 3bitri ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃 ) ↔ ( 𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒 ) )