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Mirrors > Home > MPE Home > Th. List > jad | Structured version Visualization version GIF version |
Description: Deduction form of ja 189. (Contributed by Scott Fenton, 13-Dec-2010.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
jad.1 | ⊢ (𝜑 → (¬ 𝜓 → 𝜃)) |
jad.2 | ⊢ (𝜑 → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
jad | ⊢ (𝜑 → ((𝜓 → 𝜒) → 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jad.1 | . . . 4 ⊢ (𝜑 → (¬ 𝜓 → 𝜃)) | |
2 | 1 | com12 32 | . . 3 ⊢ (¬ 𝜓 → (𝜑 → 𝜃)) |
3 | jad.2 | . . . 4 ⊢ (𝜑 → (𝜒 → 𝜃)) | |
4 | 3 | com12 32 | . . 3 ⊢ (𝜒 → (𝜑 → 𝜃)) |
5 | 2, 4 | ja 189 | . 2 ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜃)) |
6 | 5 | com12 32 | 1 ⊢ (𝜑 → ((𝜓 → 𝜒) → 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem is referenced by: pm2.6 194 pm2.65 196 merco2 1743 wereu2 5532 isfin7-2 9909 axpowndlem3 10112 suppssfz 13466 lo1bdd2 14984 pntlem3 26358 hbimtg 33369 frpomin 33396 arg-ax 34261 onsuct0 34286 ordcmp 34292 poimirlem26 35459 ax12indi 36614 ntrneiiso 41288 hbimpg 41753 |
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