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Mirrors > Home > MPE Home > Th. List > xor2 | Structured version Visualization version GIF version |
Description: Two ways to express "exclusive or". (Contributed by Mario Carneiro, 4-Sep-2016.) |
Ref | Expression |
---|---|
xor2 | ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xor 1502 | . 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) | |
2 | nbi2 1012 | . 2 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) | |
3 | 1, 2 | bitri 277 | 1 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∨ wo 843 ⊻ wxo 1501 |
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 209 df-an 399 df-or 844 df-xor 1502 |
This theorem is referenced by: xoror 1509 xornan 1510 cador 1609 saddisjlem 15813 ifpdfxor 39873 dfxor4 40131 nanorxor 40657 |
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