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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 2948 | . 2 ⊢ (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷) |
3 | 2 | necon2ai 2973 | 1 ⊢ (𝐶 = 𝐷 → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ≠ wne 2943 |
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 2944 |
This theorem is referenced by: cmpfi 22336 mcubic 25761 cubic2 25762 2sqlem11 26341 zarcmplem 31576 ovoliunnfl 35592 voliunnfl 35594 volsupnfl 35595 mncn0 40714 aaitgo 40737 |
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