Theorem syl31anc

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 )
	syl31anc.5	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )
Assertion	syl31anc	⊢ ( 𝜑 → 𝜂 )

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		syl31anc.5	⊢ ( ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜏 ) → 𝜂 )
6	1 2 3	3jca	⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
7	6 4 5	syl2anc	⊢ ( 𝜑 → 𝜂 )