Theorem breqtrd

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

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

Proof

Step	Hyp	Ref	Expression
1		breqtrd.1	⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
2		breqtrd.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
3	2	breq2d	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ 𝐴 𝑅 𝐶 ) )
4	1 3	mpbid	⊢ ( 𝜑 → 𝐴 𝑅 𝐶 )