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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 ) ) |