Metamath Proof Explorer
Description: Commuted, closed form of con3d . Proposition 30 of Frege1879 p. 44.
(Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)
|
|
Ref |
Expression |
|
Assertion |
frege30 |
⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( ¬ 𝜒 → ¬ 𝜑 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
frege29 |
⊢ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( ¬ 𝜒 → ¬ 𝜑 ) ) ) |
| 2 |
|
frege10 |
⊢ ( ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( ¬ 𝜒 → ¬ 𝜑 ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( ¬ 𝜒 → ¬ 𝜑 ) ) ) ) |
| 3 |
1 2
|
ax-mp |
⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( ¬ 𝜒 → ¬ 𝜑 ) ) ) |