Theorem weeq2

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

	Ref	Expression
Assertion	weeq2	⊢ ( 𝐴 = 𝐵 → ( 𝑅 We 𝐴 ↔ 𝑅 We 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		freq2	⊢ ( 𝐴 = 𝐵 → ( 𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵 ) )
2		soeq2	⊢ ( 𝐴 = 𝐵 → ( 𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵 ) )
3	1 2	anbi12d	⊢ ( 𝐴 = 𝐵 → ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴 ) ↔ ( 𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵 ) ) )
4		df-we	⊢ ( 𝑅 We 𝐴 ↔ ( 𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴 ) )
5		df-we	⊢ ( 𝑅 We 𝐵 ↔ ( 𝑅 Fr 𝐵 ∧ 𝑅 Or 𝐵 ) )
6	3 4 5	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( 𝑅 We 𝐴 ↔ 𝑅 We 𝐵 ) )