Theorem sylbida

Description: A syllogism deduction. (Contributed by SN, 16-Jul-2024)

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

Proof

Step	Hyp	Ref	Expression
1		sylbida.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		sylbida.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
3	1	biimpa	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
4	3 2	syldan	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )