Theorem soeq2

Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997)

	Ref	Expression
Assertion	soeq2	$$⊢A=B\to \left(R\mathrm{Or}A\leftrightarrow R\mathrm{Or}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		soss	$$⊢A\subseteq B\to \left(R\mathrm{Or}B\to R\mathrm{Or}A\right)$$
2		soss	$$⊢B\subseteq A\to \left(R\mathrm{Or}A\to R\mathrm{Or}B\right)$$
3	1 2	anim12i	$$⊢\left(A\subseteq B\wedge B\subseteq A\right)\to \left(\left(R\mathrm{Or}B\to R\mathrm{Or}A\right)\wedge \left(R\mathrm{Or}A\to R\mathrm{Or}B\right)\right)$$
4		eqss	$$⊢A=B\leftrightarrow \left(A\subseteq B\wedge B\subseteq A\right)$$
5		dfbi2	$$⊢\left(R\mathrm{Or}B\leftrightarrow R\mathrm{Or}A\right)\leftrightarrow \left(\left(R\mathrm{Or}B\to R\mathrm{Or}A\right)\wedge \left(R\mathrm{Or}A\to R\mathrm{Or}B\right)\right)$$
6	3 4 5	3imtr4i	$$⊢A=B\to \left(R\mathrm{Or}B\leftrightarrow R\mathrm{Or}A\right)$$
7	6	bicomd	$$⊢A=B\to \left(R\mathrm{Or}A\leftrightarrow R\mathrm{Or}B\right)$$