![]() |
Mathbox for Wolf Lammen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-2xor | Structured version Visualization version GIF version |
Description: In the recursive scheme
"(n+1)-xor" ↔ if-(𝜑, ¬ "n-xor" , "n-xor" ) we set n = 1 to formally arrive at an expression for "2-xor". It is based on "1-xor", that is known to be equivalent to its only input (see wl-1xor 35164). (Contributed by Wolf Lammen, 11-May-2024.) |
Ref | Expression |
---|---|
wl-2xor | ⊢ (if-(𝜑, ¬ 𝜓, 𝜓) ↔ (𝜑 ⊻ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifpdfbi 1067 | . 2 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ if-(¬ 𝜑, 𝜓, ¬ 𝜓)) | |
2 | df-xor 1504 | . . 3 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) | |
3 | nbbn 389 | . . 3 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | bitr4i 281 | . 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ (¬ 𝜑 ↔ 𝜓)) |
5 | ifpn 1070 | . 2 ⊢ (if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(¬ 𝜑, 𝜓, ¬ 𝜓)) | |
6 | 1, 4, 5 | 3bitr4ri 308 | 1 ⊢ (if-(𝜑, ¬ 𝜓, 𝜓) ↔ (𝜑 ⊻ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 if-wif 1059 ⊻ wxo 1503 |
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 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |