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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 2944 | . 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵) | |
2 | necon1bbii.1 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑) | |
3 | 1, 2 | xchnxbi 335 | 1 ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1543 ≠ wne 2940 |
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-ne 2941 |
This theorem is referenced by: necon2bbii 2992 intnex 5231 class2set 5245 csbopab 5436 relimasn 5952 modom 8879 supval2 9071 fzo0 13266 vma1 26048 lgsquadlem3 26263 ordtconnlem1 31588 |
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