Theorem syl313anc

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

	Ref	Expression
Hypotheses	syl3anc.1	⊢ ( 𝜑 → 𝜓 )
	syl3anc.2	⊢ ( 𝜑 → 𝜒 )
	syl3anc.3	⊢ ( 𝜑 → 𝜃 )
	syl3Xanc.4	⊢ ( 𝜑 → 𝜏 )
	syl23anc.5	⊢ ( 𝜑 → 𝜂 )
	syl33anc.6	⊢ ( 𝜑 → 𝜁 )
	syl133anc.7	⊢ ( 𝜑 → 𝜎 )
	syl313anc.8	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜏 ∧ ( 𝜂 ∧ 𝜁 ∧ 𝜎 ) ) → 𝜌 )
Assertion	syl313anc	⊢ ( 𝜑 → 𝜌 )

Proof

Step	Hyp	Ref	Expression
1		syl3anc.1	⊢ ( 𝜑 → 𝜓 )
2		syl3anc.2	⊢ ( 𝜑 → 𝜒 )
3		syl3anc.3	⊢ ( 𝜑 → 𝜃 )
4		syl3Xanc.4	⊢ ( 𝜑 → 𝜏 )
5		syl23anc.5	⊢ ( 𝜑 → 𝜂 )
6		syl33anc.6	⊢ ( 𝜑 → 𝜁 )
7		syl133anc.7	⊢ ( 𝜑 → 𝜎 )
8		syl313anc.8	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜏 ∧ ( 𝜂 ∧ 𝜁 ∧ 𝜎 ) ) → 𝜌 )
9	5 6 7	3jca	⊢ ( 𝜑 → ( 𝜂 ∧ 𝜁 ∧ 𝜎 ) )
10	1 2 3 4 9 8	syl311anc	⊢ ( 𝜑 → 𝜌 )