![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xorcom | Structured version Visualization version GIF version |
Description: The connector ⊻ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.) |
Ref | Expression |
---|---|
xorcom | ⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bicom 223 | . . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑)) | |
2 | 1 | notbii 321 | . 2 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ¬ (𝜓 ↔ 𝜑)) |
3 | df-xor 1497 | . 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) | |
4 | df-xor 1497 | . 2 ⊢ ((𝜓 ⊻ 𝜑) ↔ ¬ (𝜓 ↔ 𝜑)) | |
5 | 2, 3, 4 | 3bitr4i 304 | 1 ⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ⊻ wxo 1496 |
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 208 df-xor 1497 |
This theorem is referenced by: xorneg1 1507 falxortru 1569 hadcoma 1582 hadcomb 1583 cadcoma 1595 |
Copyright terms: Public domain | W3C validator |