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Mirrors > Home > MPE Home > Th. List > necon1bbid | Structured version Visualization version GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.) |
Ref | Expression |
---|---|
necon1bbid.1 | ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓)) |
Ref | Expression |
---|---|
necon1bbid | ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 3017 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | necon1bbid.1 | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓)) | |
3 | 1, 2 | syl5bbr 286 | . 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 ↔ 𝜓)) |
4 | 3 | con1bid 357 | 1 ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 = wceq 1528 ≠ 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 208 df-ne 3017 |
This theorem is referenced by: necon4abid 3056 blssioo 23332 metdstri 23388 rrxmvallem 23936 dchrpt 25771 lgsquad3 25891 eupth2lem2 27926 lkrpssN 36181 dochshpsat 38472 |
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