Theorem breqan12rd

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

	Ref	Expression
Hypotheses	breq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	breqan12i.2	⊢ ( 𝜓 → 𝐶 = 𝐷 )
Assertion	breqan12rd	⊢ ( ( 𝜓 ∧ 𝜑 ) → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		breq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		breqan12i.2	⊢ ( 𝜓 → 𝐶 = 𝐷 )
3	1 2	breqan12d	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 ) )
4	3	ancoms	⊢ ( ( 𝜓 ∧ 𝜑 ) → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐷 ) )