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 ) )