Theorem eqtr3

Description: A transitive law for class equality. (Contributed by NM, 20-May-2005)

	Ref	Expression
Assertion	eqtr3	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐶 ) → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		eqcom	⊢ ( 𝐵 = 𝐶 ↔ 𝐶 = 𝐵 )
2		eqtr	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐶 = 𝐵 ) → 𝐴 = 𝐵 )
3	1 2	sylan2b	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐶 ) → 𝐴 = 𝐵 )