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Metamath Proof Explorer

Theorem bitri

Description: An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993) (Proof shortened by Wolf Lammen, 13-Oct-2012)

	Ref	Expression
Hypotheses	bitri.1	⊢ ( 𝜑 ↔ 𝜓 )
	bitri.2	⊢ ( 𝜓 ↔ 𝜒 )
Assertion	bitri	⊢ ( 𝜑 ↔ 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		bitri.1	⊢ ( 𝜑 ↔ 𝜓 )
2		bitri.2	⊢ ( 𝜓 ↔ 𝜒 )
3	1 2	sylbb	⊢ ( 𝜑 → 𝜒 )
4	1 2	sylbbr	⊢ ( 𝜒 → 𝜑 )
5	3 4	impbii	⊢ ( 𝜑 ↔ 𝜒 )