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Mirrors > Home > MPE Home > Th. List > Mathboxes > eelT01 | 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 |
---|---|
eelT01.1 | ⊢ (⊤ → 𝜑) |
eelT01.2 | ⊢ 𝜓 |
eelT01.3 | ⊢ (𝜒 → 𝜃) |
eelT01.4 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
eelT01 | ⊢ (𝜒 → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anass 1096 | . . 3 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜒) ↔ (⊤ ∧ (𝜓 ∧ 𝜒))) | |
2 | truan 1553 | . . 3 ⊢ ((⊤ ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ 𝜒)) | |
3 | simpr 488 | . . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜒) | |
4 | eelT01.2 | . . . . 5 ⊢ 𝜓 | |
5 | 4 | jctl 527 | . . . 4 ⊢ (𝜒 → (𝜓 ∧ 𝜒)) |
6 | 3, 5 | impbii 212 | . . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ 𝜒) |
7 | 1, 2, 6 | 3bitri 300 | . 2 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜒) ↔ 𝜒) |
8 | eelT01.3 | . . 3 ⊢ (𝜒 → 𝜃) | |
9 | eelT01.1 | . . . 4 ⊢ (⊤ → 𝜑) | |
10 | eelT01.4 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) | |
11 | 9, 10 | syl3an1 1164 | . . 3 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜃) → 𝜏) |
12 | 8, 11 | syl3an3 1166 | . 2 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜒) → 𝜏) |
13 | 7, 12 | sylbir 238 | 1 ⊢ (𝜒 → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 ⊤wtru 1543 |
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 1090 df-tru 1545 |
This theorem is referenced by: (None) |
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