Theorem eqtr2

Description: A transitive law for class equality. (Contributed by NM, 20-May-2005) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 24-Oct-2024)

	Ref	Expression
Assertion	eqtr2	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐶 ) → 𝐵 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eqeq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐶 ) )
2	1	biimpa	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 = 𝐶 ) → 𝐵 = 𝐶 )