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| Mirrors > Home > MPE Home > Th. List > necon1bbii | Structured version Visualization version GIF version | ||
| Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.) | 
| Ref | Expression | 
|---|---|
| necon1bbii.1 | ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑) | 
| Ref | Expression | 
|---|---|
| necon1bbii | ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nne 2943 | . 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵) | |
| 2 | necon1bbii.1 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑) | |
| 3 | 1, 2 | xchnxbi 332 | 1 ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1539 ≠ wne 2939 | 
| 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 207 df-ne 2940 | 
| This theorem is referenced by: necon2bbii 2991 intnex 5344 class2set 5354 csbopab 5559 relimasn 6102 modom 9281 supval2 9496 fzo0 13724 vma1 27210 lgsquadlem3 27427 ordtconnlem1 33924 | 
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