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Metamath Proof Explorer

Theorem syl22anbrc

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

	Ref	Expression
Hypotheses	syl22anbrc.1	$⊢ φ \to ψ$
	syl22anbrc.2	$⊢ φ \to χ$
	syl22anbrc.3	$⊢ φ \to θ$
	syl22anbrc.4	$⊢ φ \to τ$
	syl22anbrc.5	$⊢ η \leftrightarrow ((ψ \land χ) \land (θ \land τ))$
Assertion	syl22anbrc	$⊢ φ \to η$

Step	Hyp	Ref	Expression
1		syl22anbrc.1	$⊢ φ \to ψ$
2		syl22anbrc.2	$⊢ φ \to χ$
3		syl22anbrc.3	$⊢ φ \to θ$
4		syl22anbrc.4	$⊢ φ \to τ$
5		syl22anbrc.5	$⊢ η \leftrightarrow ((ψ \land χ) \land (θ \land τ))$
6	3 4	jca	$⊢ φ \to (θ \land τ)$
7	1 2 6 5	syl21anbrc	$⊢ φ \to η$