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Mirrors > Home > MPE Home > Th. List > Mathboxes > eel0T1 | Structured version Visualization version GIF version |
Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eel0T1.1 | ⊢ 𝜑 |
eel0T1.2 | ⊢ (⊤ → 𝜓) |
eel0T1.3 | ⊢ (𝜒 → 𝜃) |
eel0T1.4 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
eel0T1 | ⊢ (𝜒 → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anass 1097 | . . 3 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜒) ↔ (𝜑 ∧ (⊤ ∧ 𝜒))) | |
2 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ (⊤ ∧ 𝜒)) → (⊤ ∧ 𝜒)) | |
3 | eel0T1.1 | . . . . 5 ⊢ 𝜑 | |
4 | 3 | jctl 527 | . . . 4 ⊢ ((⊤ ∧ 𝜒) → (𝜑 ∧ (⊤ ∧ 𝜒))) |
5 | 2, 4 | impbii 212 | . . 3 ⊢ ((𝜑 ∧ (⊤ ∧ 𝜒)) ↔ (⊤ ∧ 𝜒)) |
6 | truan 1554 | . . 3 ⊢ ((⊤ ∧ 𝜒) ↔ 𝜒) | |
7 | 1, 5, 6 | 3bitri 300 | . 2 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜒) ↔ 𝜒) |
8 | eel0T1.3 | . . 3 ⊢ (𝜒 → 𝜃) | |
9 | eel0T1.2 | . . . 4 ⊢ (⊤ → 𝜓) | |
10 | eel0T1.4 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) | |
11 | 9, 10 | syl3an2 1166 | . . 3 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜃) → 𝜏) |
12 | 8, 11 | syl3an3 1167 | . 2 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜒) → 𝜏) |
13 | 7, 12 | sylbir 238 | 1 ⊢ (𝜒 → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ⊤wtru 1544 |
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 210 df-an 400 df-3an 1091 df-tru 1546 |
This theorem is referenced by: (None) |
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