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Mirrors > Home > MPE Home > Th. List > xorneg1 | Structured version Visualization version GIF version |
Description: The connector ⊻ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 27-Jun-2020.) |
Ref | Expression |
---|---|
xorneg1 | ⊢ ((¬ 𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xorcom 1508 | . 2 ⊢ ((¬ 𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ ¬ 𝜑)) | |
2 | xorneg2 1516 | . . 3 ⊢ ((𝜓 ⊻ ¬ 𝜑) ↔ ¬ (𝜓 ⊻ 𝜑)) | |
3 | xorcom 1508 | . . 3 ⊢ ((𝜓 ⊻ 𝜑) ↔ (𝜑 ⊻ 𝜓)) | |
4 | 2, 3 | xchbinx 333 | . 2 ⊢ ((𝜓 ⊻ ¬ 𝜑) ↔ ¬ (𝜑 ⊻ 𝜓)) |
5 | 1, 4 | bitri 274 | 1 ⊢ ((¬ 𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ⊻ wxo 1505 |
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 206 df-xor 1506 |
This theorem is referenced by: xorneg 1518 |
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