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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 37982.
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 37748 is ee33VD 38137 without virtual deductions and was automatically
derived from ee33VD 38137.
|
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 74 | . . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂)))) |
5 | 4 | com4r 92 | . . . . 5 ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) |
6 | 1, 5 | syl8 74 | . . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) |
7 | pm2.43cbi 37745 | . . . . 5 ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) ↔ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) | |
8 | 7 | biimpi 205 | . . . 4 ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) |
9 | 6, 8 | e0a 38020 | . . 3 ⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) |
10 | pm2.43cbi 37745 | . . . 4 ⊢ ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) ↔ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) | |
11 | 10 | biimpi 205 | . . 3 ⊢ ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) |
12 | 9, 11 | e0a 38020 | . 2 ⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) |
13 | pm2.43cbi 37745 | . . 3 ⊢ ((𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜂)))) | |
14 | 13 | biimpi 205 | . 2 ⊢ ((𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) |
15 | 12, 14 | e0a 38020 | 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 196 |
This theorem is referenced by: (None) |
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