Description: Deduction for elimination by cases. (Contributed by NM, 8-Oct-2012)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ecased.1 | ⊢ ( 𝜑 → ( ¬ 𝜓 → 𝜃 ) ) | |
ecased.2 | ⊢ ( 𝜑 → ( ¬ 𝜒 → 𝜃 ) ) | ||
ecased.3 | ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) ) | ||
Assertion | ecased | ⊢ ( 𝜑 → 𝜃 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecased.1 | ⊢ ( 𝜑 → ( ¬ 𝜓 → 𝜃 ) ) | |
2 | ecased.2 | ⊢ ( 𝜑 → ( ¬ 𝜒 → 𝜃 ) ) | |
3 | ecased.3 | ⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) ) | |
4 | pm3.11 | ⊢ ( ¬ ( ¬ 𝜓 ∨ ¬ 𝜒 ) → ( 𝜓 ∧ 𝜒 ) ) | |
5 | 4 3 | syl5 | ⊢ ( 𝜑 → ( ¬ ( ¬ 𝜓 ∨ ¬ 𝜒 ) → 𝜃 ) ) |
6 | 1 2 5 | ecase3d | ⊢ ( 𝜑 → 𝜃 ) |