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Mirrors > Home > MPE Home > Th. List > necon2abii | Structured version Visualization version GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.) |
Ref | Expression |
---|---|
necon2abii.1 | ⊢ (𝐴 = 𝐵 ↔ ¬ 𝜑) |
Ref | Expression |
---|---|
necon2abii | ⊢ (𝜑 ↔ 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon2abii.1 | . . . 4 ⊢ (𝐴 = 𝐵 ↔ ¬ 𝜑) | |
2 | 1 | bicomi 223 | . . 3 ⊢ (¬ 𝜑 ↔ 𝐴 = 𝐵) |
3 | 2 | necon1abii 2979 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝜑) |
4 | 3 | bicomi 223 | 1 ⊢ (𝜑 ↔ 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1533 ≠ wne 2930 |
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-ne 2931 |
This theorem is referenced by: frxp2 8147 frxp3 8154 locfindis 23452 flimsncls 23908 tsmsgsum 24061 wilthlem2 27019 topdifinffinlem 36883 ismblfin 37191 elnev 43940 |
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