Theorem syl213anc

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 )
	syl213anc.7	⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ∧ ( 𝜏 ∧ 𝜂 ∧ 𝜁 ) ) → 𝜎 )
Assertion	syl213anc	⊢ ( 𝜑 → 𝜎 )

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		syl213anc.7	⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ∧ ( 𝜏 ∧ 𝜂 ∧ 𝜁 ) ) → 𝜎 )
8	1 2	jca	⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ) )
9	8 3 4 5 6 7	syl113anc	⊢ ( 𝜑 → 𝜎 )