Theorem breq1d

Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996)

	Ref	Expression
Hypothesis	breq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	breq1d	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		breq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		breq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐶 ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐶 ) )