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Mirrors > Home > MPE Home > Th. List > Mathboxes > nandsym1 | Structured version Visualization version GIF version |
Description: A symmetry with ⊼.
See negsym1 33769 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
Ref | Expression |
---|---|
nandsym1 | ⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) → (𝜓 ⊼ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nan 1482 | . . . . 5 ⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) ↔ ¬ (𝜓 ∧ (𝜓 ⊼ ⊥))) | |
2 | 1 | biimpi 218 | . . . 4 ⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) → ¬ (𝜓 ∧ (𝜓 ⊼ ⊥))) |
3 | df-nan 1482 | . . . . 5 ⊢ ((𝜓 ⊼ ⊥) ↔ ¬ (𝜓 ∧ ⊥)) | |
4 | 3 | anbi2i 624 | . . . 4 ⊢ ((𝜓 ∧ (𝜓 ⊼ ⊥)) ↔ (𝜓 ∧ ¬ (𝜓 ∧ ⊥))) |
5 | 2, 4 | sylnib 330 | . . 3 ⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) → ¬ (𝜓 ∧ ¬ (𝜓 ∧ ⊥))) |
6 | simpl 485 | . . . 4 ⊢ ((𝜓 ∧ 𝜑) → 𝜓) | |
7 | fal 1550 | . . . . 5 ⊢ ¬ ⊥ | |
8 | 7 | intnan 489 | . . . 4 ⊢ ¬ (𝜓 ∧ ⊥) |
9 | 6, 8 | jctir 523 | . . 3 ⊢ ((𝜓 ∧ 𝜑) → (𝜓 ∧ ¬ (𝜓 ∧ ⊥))) |
10 | 5, 9 | nsyl 142 | . 2 ⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) → ¬ (𝜓 ∧ 𝜑)) |
11 | df-nan 1482 | . 2 ⊢ ((𝜓 ⊼ 𝜑) ↔ ¬ (𝜓 ∧ 𝜑)) | |
12 | 10, 11 | sylibr 236 | 1 ⊢ ((𝜓 ⊼ (𝜓 ⊼ ⊥)) → (𝜓 ⊼ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ⊼ wnan 1481 ⊥wfal 1548 |
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 209 df-an 399 df-nan 1482 df-tru 1539 df-fal 1549 |
This theorem is referenced by: (None) |
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