Metamath Proof Explorer

Theorem syl223anc

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	$⊢ φ \to σ$
	syl223anc.8	$⊢ ((ψ \land χ) \land (θ \land τ) \land (η \land ζ \land σ)) \to ρ$
Assertion	syl223anc	$⊢ φ \to ρ$

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		syl223anc.8	$⊢ ((ψ \land χ) \land (θ \land τ) \land (η \land ζ \land σ)) \to ρ$
9	3 4	jca	$⊢ φ \to (θ \land τ)$
10	1 2 9 5 6 7 8	syl213anc	$⊢ φ \to ρ$