Theorem lecasei

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

	Ref	Expression
Hypotheses	lecase.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	lecase.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	lecase.3	⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → 𝜓 )
	lecase.4	⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → 𝜓 )
Assertion	lecasei	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		lecase.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		lecase.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		lecase.3	⊢ ( ( 𝜑 ∧ 𝐴 ≤ 𝐵 ) → 𝜓 )
4		lecase.4	⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → 𝜓 )
5		letric	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴 ) )
6	1 2 5	syl2anc	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴 ) )
7	3 4 6	mpjaodan	⊢ ( 𝜑 → 𝜓 )