Theorem ordeq

Description: Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993)

	Ref	Expression
Assertion	ordeq	⊢ ( 𝐴 = 𝐵 → ( Ord 𝐴 ↔ Ord 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		treq	⊢ ( 𝐴 = 𝐵 → ( Tr 𝐴 ↔ Tr 𝐵 ) )
2		weeq2	⊢ ( 𝐴 = 𝐵 → ( E We 𝐴 ↔ E We 𝐵 ) )
3	1 2	anbi12d	⊢ ( 𝐴 = 𝐵 → ( ( Tr 𝐴 ∧ E We 𝐴 ) ↔ ( Tr 𝐵 ∧ E We 𝐵 ) ) )
4		df-ord	⊢ ( Ord 𝐴 ↔ ( Tr 𝐴 ∧ E We 𝐴 ) )
5		df-ord	⊢ ( Ord 𝐵 ↔ ( Tr 𝐵 ∧ E We 𝐵 ) )
6	3 4 5	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( Ord 𝐴 ↔ Ord 𝐵 ) )