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Metamath Proof Explorer

Theorem syl3anbrc

Description: Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014)

	Ref	Expression
Hypotheses	syl3anbrc.1	$⊢ φ \to ψ$
	syl3anbrc.2	$⊢ φ \to χ$
	syl3anbrc.3	$⊢ φ \to θ$
	syl3anbrc.4	$⊢ τ \leftrightarrow (ψ \land χ \land θ)$
Assertion	syl3anbrc	$⊢ φ \to τ$

Step	Hyp	Ref	Expression
1		syl3anbrc.1	$⊢ φ \to ψ$
2		syl3anbrc.2	$⊢ φ \to χ$
3		syl3anbrc.3	$⊢ φ \to θ$
4		syl3anbrc.4	$⊢ τ \leftrightarrow (ψ \land χ \land θ)$
5	1 2 3	3jca	$⊢ φ \to (ψ \land χ \land θ)$
6	5 4	sylibr	$⊢ φ \to τ$