Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 1507 | . 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) | |
2 | nbi2 1013 | . 2 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) | |
3 | 1, 2 | bitri 274 | 1 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ 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: xoror 1514 xornan 1515 cador 1610 saddisjlem 16171 wl-df4-3mintru2 35658 ifpdfxor 41094 dfxor4 41374 nanorxor 41923 |
Copyright terms: Public domain | W3C validator |