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Mirrors > Home > MPE Home > Th. List > Mathboxes > e222 | Structured version Visualization version GIF version |
Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
e222.1 | ⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) |
e222.2 | ⊢ ( 𝜑 , 𝜓 ▶ 𝜃 ) |
e222.3 | ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) |
e222.4 | ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) |
Ref | Expression |
---|---|
e222 | ⊢ ( 𝜑 , 𝜓 ▶ 𝜂 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | e222.3 | . . . . . . 7 ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) | |
2 | 1 | dfvd2i 44583 | . . . . . 6 ⊢ (𝜑 → (𝜓 → 𝜏)) |
3 | 2 | imp 406 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜏) |
4 | e222.1 | . . . . . . . . 9 ⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) | |
5 | 4 | dfvd2i 44583 | . . . . . . . 8 ⊢ (𝜑 → (𝜓 → 𝜒)) |
6 | 5 | imp 406 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
7 | e222.2 | . . . . . . . . 9 ⊢ ( 𝜑 , 𝜓 ▶ 𝜃 ) | |
8 | 7 | dfvd2i 44583 | . . . . . . . 8 ⊢ (𝜑 → (𝜓 → 𝜃)) |
9 | 8 | imp 406 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝜃) |
10 | e222.4 | . . . . . . 7 ⊢ (𝜒 → (𝜃 → (𝜏 → 𝜂))) | |
11 | 6, 9, 10 | syl2im 40 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) → (𝜏 → 𝜂))) |
12 | 11 | pm2.43i 52 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜏 → 𝜂)) |
13 | 3, 12 | syl5com 31 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) → 𝜂)) |
14 | 13 | pm2.43i 52 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜂) |
15 | 14 | ex 412 | . 2 ⊢ (𝜑 → (𝜓 → 𝜂)) |
16 | 15 | dfvd2ir 44584 | 1 ⊢ ( 𝜑 , 𝜓 ▶ 𝜂 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ( wvd2 44575 |
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 df-vd2 44576 |
This theorem is referenced by: e220 44635 e202 44637 e022 44639 e002 44641 e020 44643 e200 44645 e221 44647 e212 44649 e122 44651 e112 44652 e121 44654 e211 44655 e22 44669 suctrALT2VD 44834 en3lplem2VD 44842 19.21a3con13vVD 44850 tratrbVD 44859 |
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