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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfxor5 | 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 |
---|---|
dfxor5 | ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((𝜑 → ¬ 𝜓) → ¬ (¬ 𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfxor4 41336 | . 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((¬ 𝜑 → 𝜓) → ¬ (𝜑 → ¬ 𝜓))) | |
2 | con2b 360 | . 2 ⊢ (((¬ 𝜑 → 𝜓) → ¬ (𝜑 → ¬ 𝜓)) ↔ ((𝜑 → ¬ 𝜓) → ¬ (¬ 𝜑 → 𝜓))) | |
3 | 1, 2 | xchbinx 334 | 1 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((𝜑 → ¬ 𝜓) → ¬ (¬ 𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ⊻ 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: (None) |
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