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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-dfnan2 | Structured version Visualization version GIF version |
Description: An alternative definition of "nand" based on imnan 437. See df-nan 1440 for the original definition. This theorem allows various shortenings. (Contributed by Wolf Lammen, 26-Jun-2020.) |
Ref | Expression |
---|---|
wl-dfnan2 | ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nan 1440 | . 2 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
2 | imnan 437 | . 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
3 | 1, 2 | bitr4i 266 | 1 ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ⊼ wnan 1439 |
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 196 df-an 385 df-nan 1440 |
This theorem is referenced by: wl-nancom 32476 wl-nannan 32477 wl-nannot 32478 wl-nanbi1 32479 wl-nanbi2 32480 |
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