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Mirrors > Home > MPE Home > Th. List > Mathboxes > ee33VD | Structured version Visualization version GIF version |
Description: Non-virtual deduction form of e33 41061.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
ee33 40848 is ee33VD 41206 without virtual deductions and was automatically
derived from ee33VD 41206.
|
Ref | Expression |
---|---|
ee33VD.1 | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) |
ee33VD.2 | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) |
ee33VD.3 | ⊢ (𝜃 → (𝜏 → 𝜂)) |
Ref | Expression |
---|---|
ee33VD | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ee33VD.2 | . . . . 5 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) | |
2 | ee33VD.1 | . . . . . . 7 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) | |
3 | ee33VD.3 | . . . . . . 7 ⊢ (𝜃 → (𝜏 → 𝜂)) | |
4 | 2, 3 | syl8 76 | . . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂)))) |
5 | 4 | com4r 94 | . . . . 5 ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) |
6 | 1, 5 | syl8 76 | . . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) |
7 | pm2.43cbi 40845 | . . . . 5 ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) ↔ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) | |
8 | 7 | biimpi 218 | . . . 4 ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) |
9 | 6, 8 | e0a 41099 | . . 3 ⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) |
10 | pm2.43cbi 40845 | . . . 4 ⊢ ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) ↔ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) | |
11 | 10 | biimpi 218 | . . 3 ⊢ ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) |
12 | 9, 11 | e0a 41099 | . 2 ⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) |
13 | pm2.43cbi 40845 | . . 3 ⊢ ((𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜂)))) | |
14 | 13 | biimpi 218 | . 2 ⊢ ((𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) |
15 | 12, 14 | e0a 41099 | 1 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 209 |
This theorem is referenced by: (None) |
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