Theorem syl323anc

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 )
	syl233anc.8	⊢ ( 𝜑 → 𝜌 )
	syl323anc.9	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ ( 𝜏 ∧ 𝜂 ) ∧ ( 𝜁 ∧ 𝜎 ∧ 𝜌 ) ) → 𝜇 )
Assertion	syl323anc	⊢ ( 𝜑 → 𝜇 )

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		syl233anc.8	⊢ ( 𝜑 → 𝜌 )
9		syl323anc.9	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ ( 𝜏 ∧ 𝜂 ) ∧ ( 𝜁 ∧ 𝜎 ∧ 𝜌 ) ) → 𝜇 )
10	4 5	jca	⊢ ( 𝜑 → ( 𝜏 ∧ 𝜂 ) )
11	1 2 3 10 6 7 8 9	syl313anc	⊢ ( 𝜑 → 𝜇 )