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Metamath Proof Explorer

Theorem syl321anc

Description: Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012)

	Ref	Expression
Hypotheses	syl3anc.1	$⊢ φ \to ψ$
	syl3anc.2	$⊢ φ \to χ$
	syl3anc.3	$⊢ φ \to θ$
	syl3Xanc.4	$⊢ φ \to τ$
	syl23anc.5	$⊢ φ \to η$
	syl33anc.6	$⊢ φ \to ζ$
	syl321anc.7	$⊢ ((ψ \land χ \land θ) \land (τ \land η) \land ζ) \to σ$
Assertion	syl321anc	$⊢ φ \to σ$

Step	Hyp	Ref	Expression
1		syl3anc.1	$⊢ φ \to ψ$
2		syl3anc.2	$⊢ φ \to χ$
3		syl3anc.3	$⊢ φ \to θ$
4		syl3Xanc.4	$⊢ φ \to τ$
5		syl23anc.5	$⊢ φ \to η$
6		syl33anc.6	$⊢ φ \to ζ$
7		syl321anc.7	$⊢ ((ψ \land χ \land θ) \land (τ \land η) \land ζ) \to σ$
8	4 5	jca	$⊢ φ \to (τ \land η)$
9	1 2 3 8 6 7	syl311anc	$⊢ φ \to σ$