Theorem breqtrrid

Description: A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005)

	Ref	Expression
Hypotheses	breqtrrid.1	⊢ 𝐴 𝑅 𝐵
	breqtrrid.2	⊢ ( 𝜑 → 𝐶 = 𝐵 )
Assertion	breqtrrid	⊢ ( 𝜑 → 𝐴 𝑅 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		breqtrrid.1	⊢ 𝐴 𝑅 𝐵
2		breqtrrid.2	⊢ ( 𝜑 → 𝐶 = 𝐵 )
3	2	eqcomd	⊢ ( 𝜑 → 𝐵 = 𝐶 )
4	1 3	breqtrid	⊢ ( 𝜑 → 𝐴 𝑅 𝐶 )