Metamath Proof Explorer
Description: Inference for elimination by cases. (Contributed by NM, 13-Jul-2005)
|
|
Ref |
Expression |
|
Hypotheses |
3ecase.1 |
⊢ ( ¬ 𝜑 → 𝜃 ) |
|
|
3ecase.2 |
⊢ ( ¬ 𝜓 → 𝜃 ) |
|
|
3ecase.3 |
⊢ ( ¬ 𝜒 → 𝜃 ) |
|
|
3ecase.4 |
⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) |
|
Assertion |
3ecase |
⊢ 𝜃 |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
3ecase.1 |
⊢ ( ¬ 𝜑 → 𝜃 ) |
2 |
|
3ecase.2 |
⊢ ( ¬ 𝜓 → 𝜃 ) |
3 |
|
3ecase.3 |
⊢ ( ¬ 𝜒 → 𝜃 ) |
4 |
|
3ecase.4 |
⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 ) |
5 |
4
|
3exp |
⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) |
6 |
1
|
2a1d |
⊢ ( ¬ 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) ) |
7 |
5 6
|
pm2.61i |
⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) ) |
8 |
7 2 3
|
pm2.61nii |
⊢ 𝜃 |