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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-1xor | Structured version Visualization version GIF version |
Description: In the recursive scheme
"(n+1)-xor" ↔ if-(𝜑, ¬ "n-xor" , "n-xor" ) we set n = 0 to formally arrive at an expression for "1-xor". The base case "0-xor" is replaced with ⊥, as a sequence of 0 inputs never has an odd number being part of it. (Contributed by Wolf Lammen, 11-May-2024.) |
Ref | Expression |
---|---|
wl-1xor | ⊢ (if-(𝜓, ¬ ⊥, ⊥) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tbtru 1547 | . . . . 5 ⊢ (𝜓 ↔ (𝜓 ↔ ⊤)) | |
2 | 1 | biimpi 219 | . . . 4 ⊢ (𝜓 → (𝜓 ↔ ⊤)) |
3 | notfal 1567 | . . . 4 ⊢ (¬ ⊥ ↔ ⊤) | |
4 | 2, 3 | bitr4di 293 | . . 3 ⊢ (𝜓 → (𝜓 ↔ ¬ ⊥)) |
5 | nbfal 1554 | . . . 4 ⊢ (¬ 𝜓 ↔ (𝜓 ↔ ⊥)) | |
6 | 5 | biimpi 219 | . . 3 ⊢ (¬ 𝜓 → (𝜓 ↔ ⊥)) |
7 | 4, 6 | casesifp 1075 | . 2 ⊢ (𝜓 ↔ if-(𝜓, ¬ ⊥, ⊥)) |
8 | 7 | bicomi 227 | 1 ⊢ (if-(𝜓, ¬ ⊥, ⊥) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 if-wif 1059 ⊤wtru 1540 ⊥wfal 1551 |
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 210 df-an 401 df-or 846 df-ifp 1060 df-tru 1542 df-fal 1552 |
This theorem is referenced by: (None) |
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