Theorem syl22anbrc

Description: Syllogism inference. (Contributed by Thierry Arnoux, 19-Oct-2025)

	Ref	Expression
Hypotheses	syl22anbrc.1	⊢ ( 𝜑 → 𝜓 )
	syl22anbrc.2	⊢ ( 𝜑 → 𝜒 )
	syl22anbrc.3	⊢ ( 𝜑 → 𝜃 )
	syl22anbrc.4	⊢ ( 𝜑 → 𝜏 )
	syl22anbrc.5	⊢ ( 𝜂 ↔ ( ( 𝜓 ∧ 𝜒 ) ∧ ( 𝜃 ∧ 𝜏 ) ) )
Assertion	syl22anbrc	⊢ ( 𝜑 → 𝜂 )

Proof

Step	Hyp	Ref	Expression
1		syl22anbrc.1	⊢ ( 𝜑 → 𝜓 )
2		syl22anbrc.2	⊢ ( 𝜑 → 𝜒 )
3		syl22anbrc.3	⊢ ( 𝜑 → 𝜃 )
4		syl22anbrc.4	⊢ ( 𝜑 → 𝜏 )
5		syl22anbrc.5	⊢ ( 𝜂 ↔ ( ( 𝜓 ∧ 𝜒 ) ∧ ( 𝜃 ∧ 𝜏 ) ) )
6	3 4	jca	⊢ ( 𝜑 → ( 𝜃 ∧ 𝜏 ) )
7	1 2 6 5	syl21anbrc	⊢ ( 𝜑 → 𝜂 )