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Mirrors > Home > MPE Home > Th. List > norcom | Structured version Visualization version GIF version |
Description: The connector ⊽ is commutative. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 23-Apr-2024.) |
Ref | Expression |
---|---|
norcom | ⊢ ((𝜑 ⊽ 𝜓) ↔ (𝜓 ⊽ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nor 1525 | . . 3 ⊢ ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓)) | |
2 | orcom 866 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑)) | |
3 | 1, 2 | xchbinx 333 | . 2 ⊢ ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜓 ∨ 𝜑)) |
4 | df-nor 1525 | . 2 ⊢ ((𝜓 ⊽ 𝜑) ↔ ¬ (𝜓 ∨ 𝜑)) | |
5 | 3, 4 | bitr4i 277 | 1 ⊢ ((𝜑 ⊽ 𝜓) ↔ (𝜓 ⊽ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∨ wo 843 ⊽ wnor 1524 |
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-or 844 df-nor 1525 |
This theorem is referenced by: norass 1537 norassOLD 1538 falnortru 1594 |
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