![]() |
Mathbox for Wolf Lammen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-df3xor2 | Structured version Visualization version GIF version |
Description: Alternative definition of wl-df-3xor 35150, using triple exclusive disjunction, or XOR3. You can add more input by appending each one with a ⊻. Copy of hadass 1599. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 1-May-2024.) |
Ref | Expression |
---|---|
wl-df3xor2 | ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifpn 1070 | . 2 ⊢ (if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒)) ↔ if-(¬ 𝜑, (𝜓 ⊻ 𝜒), ¬ (𝜓 ⊻ 𝜒))) | |
2 | wl-df-3xor 35150 | . 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒))) | |
3 | df-xor 1504 | . . 3 ⊢ ((𝜑 ⊻ (𝜓 ⊻ 𝜒)) ↔ ¬ (𝜑 ↔ (𝜓 ⊻ 𝜒))) | |
4 | nbbn 389 | . . 3 ⊢ ((¬ 𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ ¬ (𝜑 ↔ (𝜓 ⊻ 𝜒))) | |
5 | ifpdfbi 1067 | . . 3 ⊢ ((¬ 𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ if-(¬ 𝜑, (𝜓 ⊻ 𝜒), ¬ (𝜓 ⊻ 𝜒))) | |
6 | 3, 4, 5 | 3bitr2i 303 | . 2 ⊢ ((𝜑 ⊻ (𝜓 ⊻ 𝜒)) ↔ if-(¬ 𝜑, (𝜓 ⊻ 𝜒), ¬ (𝜓 ⊻ 𝜒))) |
7 | 1, 2, 6 | 3bitr4i 307 | 1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ 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: wl-df3xor3 35152 wl-3xorbi 35155 |
Copyright terms: Public domain | W3C validator |