Theorem freq2

Description: Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994)

	Ref	Expression
Assertion	freq2	⊢ ( 𝐴 = 𝐵 → ( 𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eqimss2	⊢ ( 𝐴 = 𝐵 → 𝐵 ⊆ 𝐴 )
2		frss	⊢ ( 𝐵 ⊆ 𝐴 → ( 𝑅 Fr 𝐴 → 𝑅 Fr 𝐵 ) )
3	1 2	syl	⊢ ( 𝐴 = 𝐵 → ( 𝑅 Fr 𝐴 → 𝑅 Fr 𝐵 ) )
4		eqimss	⊢ ( 𝐴 = 𝐵 → 𝐴 ⊆ 𝐵 )
5		frss	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑅 Fr 𝐵 → 𝑅 Fr 𝐴 ) )
6	4 5	syl	⊢ ( 𝐴 = 𝐵 → ( 𝑅 Fr 𝐵 → 𝑅 Fr 𝐴 ) )
7	3 6	impbid	⊢ ( 𝐴 = 𝐵 → ( 𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵 ) )