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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 628), 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 1265 | . 2 wff (φ ⊻ ψ) |
4 | 1, 2 | wo 628 | . . 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 1267 xorbi2d 1269 xorbi1d 1270 xorbin 1272 xorcom 1276 xornbidc 1279 xordc1 1281 anxordi 1288 truxortru 1307 truxorfal 1308 falxortru 1309 falxorfal 1310 mpto2 1312 mtp-xor 1313 reapltxor 7373 bdxor 9291 |
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