Theorem breqtrrd

Description: Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999)

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

Proof

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