Theorem weeq2

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

	Ref	Expression
Assertion	weeq2	$$⊢A=B\to \left(R\mathrm{We}A\leftrightarrow R\mathrm{We}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		freq2	$$⊢A=B\to \left(R\mathrm{Fr}A\leftrightarrow R\mathrm{Fr}B\right)$$
2		soeq2	$$⊢A=B\to \left(R\mathrm{Or}A\leftrightarrow R\mathrm{Or}B\right)$$
3	1 2	anbi12d	$$⊢A=B\to \left(\left(R\mathrm{Fr}A\wedge R\mathrm{Or}A\right)\leftrightarrow \left(R\mathrm{Fr}B\wedge R\mathrm{Or}B\right)\right)$$
4		df-we	$$⊢R\mathrm{We}A\leftrightarrow \left(R\mathrm{Fr}A\wedge R\mathrm{Or}A\right)$$
5		df-we	$$⊢R\mathrm{We}B\leftrightarrow \left(R\mathrm{Fr}B\wedge R\mathrm{Or}B\right)$$
6	3 4 5	3bitr4g	$$⊢A=B\to \left(R\mathrm{We}A\leftrightarrow R\mathrm{We}B\right)$$