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Metamath Proof Explorer

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 )