Theorem anass

Description: Associative law for conjunction. Theorem *4.32 of WhiteheadRussell p. 118. (Contributed by NM, 21-Jun-1993) (Proof shortened by Wolf Lammen, 24-Nov-2012)

	Ref	Expression
Assertion	anass	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) → ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) )
2	1	anassrs	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) )
3		id	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
4	3	anasss	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) → ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
5	2 4	impbii	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) )