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Mirrors > Home > MPE Home > Th. List > necon1abid | Structured version Visualization version GIF version |
Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.) |
Ref | Expression |
---|---|
necon1abid.1 | ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵)) |
Ref | Expression |
---|---|
necon1abid | ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotb 304 | . 2 ⊢ (𝜓 ↔ ¬ ¬ 𝜓) | |
2 | necon1abid.1 | . . 3 ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 = 𝐵)) | |
3 | 2 | necon3bbid 2831 | . 2 ⊢ (𝜑 → (¬ ¬ 𝜓 ↔ 𝐴 ≠ 𝐵)) |
4 | 1, 3 | syl5rbb 273 | 1 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 = wceq 1483 ≠ wne 2794 |
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 197 df-ne 2795 |
This theorem is referenced by: lttri2 10120 xrlttri2 11975 ioon0 12201 lssne0 18951 xmetgt0 22163 sotrine 31658 |
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