Metamath Proof Explorer
Theorem jad
Description: Deduction form of ja . (Contributed by Scott Fenton, 13-Dec-2010)
(Proof shortened by Andrew Salmon, 17-Sep-2011)
|
|
Ref |
Expression |
|
Hypotheses |
jad.1 |
⊢ ( 𝜑 → ( ¬ 𝜓 → 𝜃 ) ) |
|
|
jad.2 |
⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) ) |
|
Assertion |
jad |
⊢ ( 𝜑 → ( ( 𝜓 → 𝜒 ) → 𝜃 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
jad.1 |
⊢ ( 𝜑 → ( ¬ 𝜓 → 𝜃 ) ) |
2 |
|
jad.2 |
⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) ) |
3 |
1
|
com12 |
⊢ ( ¬ 𝜓 → ( 𝜑 → 𝜃 ) ) |
4 |
2
|
com12 |
⊢ ( 𝜒 → ( 𝜑 → 𝜃 ) ) |
5 |
3 4
|
ja |
⊢ ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜃 ) ) |
6 |
5
|
com12 |
⊢ ( 𝜑 → ( ( 𝜓 → 𝜒 ) → 𝜃 ) ) |