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Mirrors > Home > MPE Home > Th. List > Mathboxes > plvcofphax | Structured version Visualization version GIF version |
Description: Given, a,b,d, and "definitions" for c, e, f, g: g is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.) |
Ref | Expression |
---|---|
plvcofphax.1 | ⊢ (𝜒 ↔ ((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑)))) |
plvcofphax.2 | ⊢ (𝜏 ↔ ((𝜒 → 𝜃) ∧ (𝜑 ↔ 𝜒) ∧ ((𝜑 → 𝜓) → (𝜓 ↔ 𝜃)))) |
plvcofphax.3 | ⊢ (𝜂 ↔ (𝜒 ∧ 𝜏)) |
plvcofphax.4 | ⊢ 𝜑 |
plvcofphax.5 | ⊢ 𝜓 |
plvcofphax.6 | ⊢ 𝜃 |
plvcofphax.7 | ⊢ (𝜁 ↔ ¬ (𝜓 ∧ ¬ 𝜏)) |
Ref | Expression |
---|---|
plvcofphax | ⊢ 𝜁 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plvcofphax.5 | . . . . 5 ⊢ 𝜓 | |
2 | plvcofphax.2 | . . . . . 6 ⊢ (𝜏 ↔ ((𝜒 → 𝜃) ∧ (𝜑 ↔ 𝜒) ∧ ((𝜑 → 𝜓) → (𝜓 ↔ 𝜃)))) | |
3 | plvcofphax.4 | . . . . . 6 ⊢ 𝜑 | |
4 | plvcofphax.1 | . . . . . . 7 ⊢ (𝜒 ↔ ((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑)))) | |
5 | 4, 3, 1 | plcofph 44111 | . . . . . 6 ⊢ 𝜒 |
6 | plvcofphax.6 | . . . . . 6 ⊢ 𝜃 | |
7 | 2, 3, 1, 5, 6 | pldofph 44112 | . . . . 5 ⊢ 𝜏 |
8 | 1, 7 | pm3.2i 474 | . . . 4 ⊢ (𝜓 ∧ 𝜏) |
9 | pm3.4 810 | . . . 4 ⊢ ((𝜓 ∧ 𝜏) → (𝜓 → 𝜏)) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ (𝜓 → 𝜏) |
11 | iman 405 | . . . 4 ⊢ ((𝜓 → 𝜏) ↔ ¬ (𝜓 ∧ ¬ 𝜏)) | |
12 | 11 | biimpi 219 | . . 3 ⊢ ((𝜓 → 𝜏) → ¬ (𝜓 ∧ ¬ 𝜏)) |
13 | 10, 12 | ax-mp 5 | . 2 ⊢ ¬ (𝜓 ∧ ¬ 𝜏) |
14 | plvcofphax.7 | . . . 4 ⊢ (𝜁 ↔ ¬ (𝜓 ∧ ¬ 𝜏)) | |
15 | 14 | bicomi 227 | . . 3 ⊢ (¬ (𝜓 ∧ ¬ 𝜏) ↔ 𝜁) |
16 | 15 | biimpi 219 | . 2 ⊢ (¬ (𝜓 ∧ ¬ 𝜏) → 𝜁) |
17 | 13, 16 | ax-mp 5 | 1 ⊢ 𝜁 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 |
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 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |