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Mirrors > Home > MPE Home > Th. List > necon2i | Structured version Visualization version GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) |
Ref | Expression |
---|---|
necon2i.1 | ⊢ (𝐴 = 𝐵 → 𝐶 ≠ 𝐷) |
Ref | Expression |
---|---|
necon2i | ⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon2i.1 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐶 ≠ 𝐷) | |
2 | 1 | neneqd 3021 | . 2 ⊢ (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷) |
3 | 2 | necon2ai 3045 | 1 ⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ≠ wne 3016 |
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 209 df-ne 3017 |
This theorem is referenced by: cmpfi 22015 mcubic 25424 cubic2 25425 2sqlem11 26004 ovoliunnfl 34933 voliunnfl 34935 volsupnfl 34936 mncn0 39737 aaitgo 39760 |
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