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Metamath Proof Explorer
Description: An inference for implication elimination. (Contributed by Giovanni
Mascellani, 23-May-2019) (Proof shortened by Wolf Lammen, 2-Sep-2020)
|
|
Ref |
Expression |
|
Hypotheses |
impel.1 |
⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) |
|
|
impel.2 |
⊢ ( 𝜃 → 𝜓 ) |
|
Assertion |
impel |
⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜒 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
impel.1 |
⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) ) |
| 2 |
|
impel.2 |
⊢ ( 𝜃 → 𝜓 ) |
| 3 |
2 1
|
syl5 |
⊢ ( 𝜑 → ( 𝜃 → 𝜒 ) ) |
| 4 |
3
|
imp |
⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜒 ) |