Theorem weeq1

Description: Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997)

	Ref	Expression
Assertion	weeq1	⊢ ( 𝑅 = 𝑆 → ( 𝑅 We 𝐴 ↔ 𝑆 We 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		freq1	⊢ ( 𝑅 = 𝑆 → ( 𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴 ) )
2		soeq1	⊢ ( 𝑅 = 𝑆 → ( 𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴 ) )
3	1 2	anbi12d	⊢ ( 𝑅 = 𝑆 → ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴 ) ↔ ( 𝑆 Fr 𝐴 ∧ 𝑆 Or 𝐴 ) ) )
4		df-we	⊢ ( 𝑅 We 𝐴 ↔ ( 𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴 ) )
5		df-we	⊢ ( 𝑆 We 𝐴 ↔ ( 𝑆 Fr 𝐴 ∧ 𝑆 Or 𝐴 ) )
6	3 4 5	3bitr4g	⊢ ( 𝑅 = 𝑆 → ( 𝑅 We 𝐴 ↔ 𝑆 We 𝐴 ) )