Theorem syl21anbrc

Description: Syllogism inference. (Contributed by Peter Mazsa, 18-Sep-2022)

	Ref	Expression
Hypotheses	syl21anbrc.1	⊢ ( 𝜑 → 𝜓 )
	syl21anbrc.2	⊢ ( 𝜑 → 𝜒 )
	syl21anbrc.3	⊢ ( 𝜑 → 𝜃 )
	syl21anbrc.4	⊢ ( 𝜏 ↔ ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) )
Assertion	syl21anbrc	⊢ ( 𝜑 → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		syl21anbrc.1	⊢ ( 𝜑 → 𝜓 )
2		syl21anbrc.2	⊢ ( 𝜑 → 𝜒 )
3		syl21anbrc.3	⊢ ( 𝜑 → 𝜃 )
4		syl21anbrc.4	⊢ ( 𝜏 ↔ ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) )
5	1 2 3	jca31	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) )
6	5 4	sylibr	⊢ ( 𝜑 → 𝜏 )