Theorem eqbrtrrd

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

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

Proof

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