Theorem sylanblc

Description: Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019)

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

Proof

Step	Hyp	Ref	Expression
1		sylanblc.1	⊢ ( 𝜑 → 𝜓 )
2		sylanblc.2	⊢ 𝜒
3		sylanblc.3	⊢ ( ( 𝜓 ∧ 𝜒 ) ↔ 𝜃 )
4	3	biimpi	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
5	1 2 4	sylancl	⊢ ( 𝜑 → 𝜃 )