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Mirrors > Home > MPE Home > Th. List > pm4.82 | Structured version Visualization version GIF version |
Description: Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) |
Ref | Expression |
---|---|
pm4.82 | ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.65 196 | . . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑)) | |
2 | 1 | imp 410 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) → ¬ 𝜑) |
3 | pm2.21 123 | . . 3 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
4 | pm2.21 123 | . . 3 ⊢ (¬ 𝜑 → (𝜑 → ¬ 𝜓)) | |
5 | 3, 4 | jca 515 | . 2 ⊢ (¬ 𝜑 → ((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓))) |
6 | 2, 5 | impbii 212 | 1 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 |
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 |
This theorem is referenced by: alimp-no-surprise 45938 |
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