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Mirrors > Home > MPE Home > Th. List > dfnan2 | Structured version Visualization version GIF version |
Description: Alternative denial in terms of our primitive connectives (implication and negation). (Contributed by WL, 26-Jun-2020.) |
Ref | Expression |
---|---|
dfnan2 | ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nan 1488 | . 2 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
2 | imnan 403 | . 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
3 | 1, 2 | bitr4i 281 | 1 ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ⊼ wnan 1487 |
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-nan 1488 |
This theorem is referenced by: nancom 1492 nannan 1493 nannot 1495 nanbi1 1497 |
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