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Mirrors > Home > MPE Home > Th. List > Mathboxes > eelTT1 | 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 |
---|---|
eelTT1.1 | ⊢ (⊤ → 𝜑) |
eelTT1.2 | ⊢ (⊤ → 𝜓) |
eelTT1.3 | ⊢ (𝜒 → 𝜃) |
eelTT1.4 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
eelTT1 | ⊢ (𝜒 → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anass 1035 | . . 3 ⊢ ((⊤ ∧ ⊤ ∧ 𝜒) ↔ (⊤ ∧ (⊤ ∧ 𝜒))) | |
2 | anabs5 847 | . . 3 ⊢ ((⊤ ∧ (⊤ ∧ 𝜒)) ↔ (⊤ ∧ 𝜒)) | |
3 | truan 1492 | . . 3 ⊢ ((⊤ ∧ 𝜒) ↔ 𝜒) | |
4 | 1, 2, 3 | 3bitri 285 | . 2 ⊢ ((⊤ ∧ ⊤ ∧ 𝜒) ↔ 𝜒) |
5 | eelTT1.3 | . . 3 ⊢ (𝜒 → 𝜃) | |
6 | eelTT1.2 | . . . 4 ⊢ (⊤ → 𝜓) | |
7 | eelTT1.1 | . . . . 5 ⊢ (⊤ → 𝜑) | |
8 | eelTT1.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) | |
9 | 7, 8 | syl3an1 1351 | . . . 4 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜃) → 𝜏) |
10 | 6, 9 | syl3an2 1352 | . . 3 ⊢ ((⊤ ∧ ⊤ ∧ 𝜃) → 𝜏) |
11 | 5, 10 | syl3an3 1353 | . 2 ⊢ ((⊤ ∧ ⊤ ∧ 𝜒) → 𝜏) |
12 | 4, 11 | sylbir 224 | 1 ⊢ (𝜒 → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ⊤wtru 1476 |
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 df-an 385 df-3an 1033 df-tru 1478 |
This theorem is referenced by: (None) |
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