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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfxor4 | Structured version Visualization version GIF version |
Description: Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.) |
Ref | Expression |
---|---|
dfxor4 | ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((¬ 𝜑 → 𝜓) → ¬ (𝜑 → ¬ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xor2 1513 | . 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) | |
2 | df-or 845 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | |
3 | imnan 400 | . . . 4 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
4 | 3 | bicomi 223 | . . 3 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (𝜑 → ¬ 𝜓)) |
5 | 2, 4 | anbi12i 627 | . 2 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ ((¬ 𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓))) |
6 | df-an 397 | . 2 ⊢ (((¬ 𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ ((¬ 𝜑 → 𝜓) → ¬ (𝜑 → ¬ 𝜓))) | |
7 | 1, 5, 6 | 3bitri 297 | 1 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((¬ 𝜑 → 𝜓) → ¬ (𝜑 → ¬ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ⊻ wxo 1506 |
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 397 df-or 845 df-xor 1507 |
This theorem is referenced by: dfxor5 41345 |
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