Theorem syl3anbr

Description: A triple syllogism inference. (Contributed by NM, 29-Dec-2011)

	Ref	Expression
Hypotheses	syl3anbr.1	⊢ ( 𝜓 ↔ 𝜑 )
	syl3anbr.2	⊢ ( 𝜃 ↔ 𝜒 )
	syl3anbr.3	⊢ ( 𝜂 ↔ 𝜏 )
	syl3anbr.4	⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) → 𝜁 )
Assertion	syl3anbr	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜁 )

Proof

Step	Hyp	Ref	Expression
1		syl3anbr.1	⊢ ( 𝜓 ↔ 𝜑 )
2		syl3anbr.2	⊢ ( 𝜃 ↔ 𝜒 )
3		syl3anbr.3	⊢ ( 𝜂 ↔ 𝜏 )
4		syl3anbr.4	⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) → 𝜁 )
5	1	bicomi	⊢ ( 𝜑 ↔ 𝜓 )
6	2	bicomi	⊢ ( 𝜒 ↔ 𝜃 )
7	3	bicomi	⊢ ( 𝜏 ↔ 𝜂 )
8	5 6 7 4	syl3anb	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜁 )