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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-3xorfal | Structured version Visualization version GIF version |
Description: If the first input is false, then triple xor is equivalent to the exclusive disjunction of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 29-Apr-2024.) |
Ref | Expression |
---|---|
wl-3xorfal | ⊢ (¬ 𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ⊻ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-df-3xor 35150 | . 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒))) | |
2 | ifpfal 1073 | . 2 ⊢ (¬ 𝜑 → (if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒)) ↔ (𝜓 ⊻ 𝜒))) | |
3 | 1, 2 | syl5bb 286 | 1 ⊢ (¬ 𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ⊻ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 if-wif 1059 ⊻ wxo 1503 haddwhad 1595 |
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-xor 1504 df-tru 1542 df-had 1596 |
This theorem is referenced by: (None) |
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