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Mathbox for Alan Sare |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ee222 | Structured version Visualization version GIF version |
Description: e222 44634 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ee222.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
ee222.2 | ⊢ (𝜑 → (𝜓 → 𝜃)) |
ee222.3 | ⊢ (𝜑 → (𝜓 → 𝜏)) |
ee222.4 | ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) |
Ref | Expression |
---|---|
ee222 | ⊢ (𝜑 → (𝜓 → 𝜂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ee222.1 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | imp 406 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
3 | ee222.2 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜃)) | |
4 | 3 | imp 406 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃) |
5 | ee222.3 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜏)) | |
6 | 5 | imp 406 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜏) |
7 | ee222.4 | . . 3 ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) | |
8 | 2, 4, 6, 7 | syl3c 66 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜂) |
9 | 8 | ex 412 | 1 ⊢ (𝜑 → (𝜓 → 𝜂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 |
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 207 df-an 396 |
This theorem is referenced by: ee121 44503 ee122 44504 tratrb 44534 ee220 44636 ee202 44638 ee022 44640 ee002 44642 ee020 44644 ee200 44646 ee221 44648 ee212 44650 ee112 44653 ee211 44656 ee210 44658 ee201 44660 ee120 44662 ee021 44664 ee012 44666 ee102 44668 suctrALT2 44835 |
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