Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-nancom | Structured version Visualization version GIF version |
Description: The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.) (Revised by Wolf Lammen, 26-Jun-2020.) |
Ref | Expression |
---|---|
wl-nancom | ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con2b 349 | . 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑)) | |
2 | wl-dfnan2 33296 | . 2 ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓)) | |
3 | wl-dfnan2 33296 | . 2 ⊢ ((𝜓 ⊼ 𝜑) ↔ (𝜓 → ¬ 𝜑)) | |
4 | 1, 2, 3 | 3bitr4i 292 | 1 ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ⊼ wnan 1447 |
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 197 df-an 386 df-nan 1448 |
This theorem is referenced by: (None) |
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