Theorem syl22anc

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

	Ref	Expression
Hypotheses	syl12anc.1	⊢ ( 𝜑 → 𝜓 )
	syl12anc.2	⊢ ( 𝜑 → 𝜒 )
	syl12anc.3	⊢ ( 𝜑 → 𝜃 )
	syl22anc.4	⊢ ( 𝜑 → 𝜏 )
	syl22anc.5	⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ ( 𝜃 ∧ 𝜏 ) ) → 𝜂 )
Assertion	syl22anc	⊢ ( 𝜑 → 𝜂 )

Proof

Step	Hyp	Ref	Expression
1		syl12anc.1	⊢ ( 𝜑 → 𝜓 )
2		syl12anc.2	⊢ ( 𝜑 → 𝜒 )
3		syl12anc.3	⊢ ( 𝜑 → 𝜃 )
4		syl22anc.4	⊢ ( 𝜑 → 𝜏 )
5		syl22anc.5	⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ ( 𝜃 ∧ 𝜏 ) ) → 𝜂 )
6	1 2	jca	⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ) )
7	6 3 4 5	syl12anc	⊢ ( 𝜑 → 𝜂 )