Theorem releq

Description: Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994)

	Ref	Expression
Assertion	releq	⊢ ( 𝐴 = 𝐵 → ( Rel 𝐴 ↔ Rel 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sseq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⊆ ( V × V ) ↔ 𝐵 ⊆ ( V × V ) ) )
2		df-rel	⊢ ( Rel 𝐴 ↔ 𝐴 ⊆ ( V × V ) )
3		df-rel	⊢ ( Rel 𝐵 ↔ 𝐵 ⊆ ( V × V ) )
4	1 2 3	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( Rel 𝐴 ↔ Rel 𝐵 ) )