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Mirrors > Home > MPE Home > Th. List > noran | Structured version Visualization version GIF version |
Description: ∧ is expressible via ⊽. (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.) |
Ref | Expression |
---|---|
noran | ⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anor 979 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓)) | |
2 | nornot 1528 | . . . 4 ⊢ (¬ 𝜑 ↔ (𝜑 ⊽ 𝜑)) | |
3 | nornot 1528 | . . . 4 ⊢ (¬ 𝜓 ↔ (𝜓 ⊽ 𝜓)) | |
4 | 2, 3 | orbi12i 911 | . . 3 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ((𝜑 ⊽ 𝜑) ∨ (𝜓 ⊽ 𝜓))) |
5 | 1, 4 | xchbinx 333 | . 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ ((𝜑 ⊽ 𝜑) ∨ (𝜓 ⊽ 𝜓))) |
6 | df-nor 1525 | . 2 ⊢ (((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓)) ↔ ¬ ((𝜑 ⊽ 𝜑) ∨ (𝜓 ⊽ 𝜓))) | |
7 | 5, 6 | bitr4i 277 | 1 ⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 ∨ wo 843 ⊽ wnor 1524 |
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 206 df-an 396 df-or 844 df-nor 1525 |
This theorem is referenced by: (None) |
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