Metamath Proof Explorer

Theorem syl322anc

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-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	$⊢ φ \to ζ$
	syl133anc.7	$⊢ φ \to σ$
	syl322anc.8	$⊢ ((ψ \land χ \land θ) \land (τ \land η) \land (ζ \land σ)) \to ρ$
Assertion	syl322anc	$⊢ φ \to ρ$

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		syl133anc.7	$⊢ φ \to σ$
8		syl322anc.8	$⊢ ((ψ \land χ \land θ) \land (τ \land η) \land (ζ \land σ)) \to ρ$
9	6 7	jca	$⊢ φ \to (ζ \land σ)$
10	1 2 3 4 5 9 8	syl321anc	$⊢ φ \to ρ$