Theorem lecasei

Description: Ordering elimination by cases. (Contributed by NM, 6-Jul-2007)

	Ref	Expression
Hypotheses	lecase.1	$$⊢\phi \to A\in ℝ$$
	lecase.2	$$⊢\phi \to B\in ℝ$$
	lecase.3	$$⊢\left(\phi \wedge A\le B\right)\to \psi$$
	lecase.4	$$⊢\left(\phi \wedge B\le A\right)\to \psi$$
Assertion	lecasei	$$⊢\phi \to \psi$$

Proof

Step	Hyp	Ref	Expression
1		lecase.1	$$⊢\phi \to A\in ℝ$$
2		lecase.2	$$⊢\phi \to B\in ℝ$$
3		lecase.3	$$⊢\left(\phi \wedge A\le B\right)\to \psi$$
4		lecase.4	$$⊢\left(\phi \wedge B\le A\right)\to \psi$$
5		letric	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A\le B\vee B\le A\right)$$
6	1 2 5	syl2anc	$$⊢\phi \to \left(A\le B\vee B\le A\right)$$
7	3 4 6	mpjaodan	$$⊢\phi \to \psi$$