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Metamath Proof Explorer

Theorem ancom1s

Description: Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006) (Proof shortened by Wolf Lammen, 24-Nov-2012)

		Ref	Expression
	Hypothesis	an32s.1	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 )
	Assertion	ancom1s	⊢ ( ( ( 𝜓 ∧ 𝜑 ) ∧ 𝜒 ) → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		an32s.1	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) → 𝜃 )
2		pm3.22	⊢ ( ( 𝜓 ∧ 𝜑 ) → ( 𝜑 ∧ 𝜓 ) )
3	2 1	sylan	⊢ ( ( ( 𝜓 ∧ 𝜑 ) ∧ 𝜒 ) → 𝜃 )