Theorem syl21anc

Description: Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009)

	Ref	Expression
Hypotheses	syl12anc.1	⊢ ( 𝜑 → 𝜓 )
	syl12anc.2	⊢ ( 𝜑 → 𝜒 )
	syl12anc.3	⊢ ( 𝜑 → 𝜃 )
	syl21anc.4	⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )
Assertion	syl21anc	⊢ ( 𝜑 → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		syl12anc.1	⊢ ( 𝜑 → 𝜓 )
2		syl12anc.2	⊢ ( 𝜑 → 𝜒 )
3		syl12anc.3	⊢ ( 𝜑 → 𝜃 )
4		syl21anc.4	⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )
5	1 2	jca	⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ) )
6	5 3 4	syl2anc	⊢ ( 𝜑 → 𝜏 )