Theorem breq12d

Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996) (Proof shortened by Andrew Salmon, 9-Jul-2011)

	Ref	Expression
Hypotheses	breq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	breq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	breq12d	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		breq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		breq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3		breq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 ) )