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| Mirrors > Home > MPE Home > Th. List > xor3 | Structured version Visualization version GIF version | ||
| Description: Two ways to express "exclusive or". (Contributed by NM, 1-Jan-2006.) |
| Ref | Expression |
|---|---|
| xor3 | ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm5.18 381 | . . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)) | |
| 2 | 1 | con2bii 357 | . 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) |
| 3 | 2 | bicomi 224 | 1 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 |
| 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 207 |
| This theorem is referenced by: nbbn 383 pm5.15 1015 nbi2 1018 xorass 1515 hadnot 1602 nabbib 3045 nmogtmnf 30789 nmopgtmnf 31887 limsucncmpi 36446 wl-3xorbi 37474 wl-3xornot 37482 oneptri 43269 oaordnrex 43308 omnord1ex 43317 oenord1ex 43328 aiffnbandciffatnotciffb 46916 axorbciffatcxorb 46917 abnotbtaxb 46927 afv2orxorb 47240 line2ylem 48672 line2xlem 48674 |
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