Theorem syl2anc2

Description: Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023)

	Ref	Expression
Hypotheses	syl2anc2.1	⊢ ( 𝜑 → 𝜓 )
	syl2anc2.2	⊢ ( 𝜓 → 𝜒 )
	syl2anc2.3	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
Assertion	syl2anc2	⊢ ( 𝜑 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		syl2anc2.1	⊢ ( 𝜑 → 𝜓 )
2		syl2anc2.2	⊢ ( 𝜓 → 𝜒 )
3		syl2anc2.3	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
4	1 2	syl	⊢ ( 𝜑 → 𝜒 )
5	1 4 3	syl2anc	⊢ ( 𝜑 → 𝜃 )