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Mirrors > Home > ILE Home > Th. List > df-xor | GIF version |
Description: Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with ∧ (wa 97), ∨ (wo 629), and → (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.) |
Ref | Expression |
---|---|
df-xor | ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | wps | . . 3 wff 𝜓 | |
3 | 1, 2 | wxo 1266 | . 2 wff (𝜑 ⊻ 𝜓) |
4 | 1, 2 | wo 629 | . . 3 wff (𝜑 ∨ 𝜓) |
5 | 1, 2 | wa 97 | . . . 4 wff (𝜑 ∧ 𝜓) |
6 | 5 | wn 3 | . . 3 wff ¬ (𝜑 ∧ 𝜓) |
7 | 4, 6 | wa 97 | . 2 wff ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) |
8 | 3, 7 | wb 98 | 1 wff ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) |
Colors of variables: wff set class |
This definition is referenced by: xoranor 1268 xorbi2d 1271 xorbi1d 1272 xorbin 1275 xorcom 1279 xornbidc 1282 xordc1 1284 anxordi 1291 truxortru 1310 truxorfal 1311 falxortru 1312 falxorfal 1313 mptxor 1315 reapltxor 7580 bdxor 9956 |
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