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Metamath Proof Explorer

Theorem 3anan12

Description: Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (Proof shortened by Andrew Salmon, 14-Jun-2011) (Revised to shorten 3ancoma by Wolf Lammen, 5-Jun-2022.)

		Ref	Expression
	Assertion	3anan12	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( 𝜓 ∧ ( 𝜑 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		3anass	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) )
2		an12	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) ↔ ( 𝜓 ∧ ( 𝜑 ∧ 𝜒 ) ) )
3	1 2	bitri	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( 𝜓 ∧ ( 𝜑 ∧ 𝜒 ) ) )