Theorem 3anasss

Description: Associative law for conjunction applied to antecedent (eliminates syllogism). Converse of 3anassrs . (Contributed by Thierry Arnoux, 5-Jul-2026)

	Ref	Expression
Hypothesis	3anasss.1	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )
Assertion	3anasss	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		3anasss.1	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )
2		13an22anass	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ) )
3	1	anasss	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ) → 𝜏 )
4	2 3	sylbi	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ) → 𝜏 )