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| Mirrors > Home > MPE Home > Th. List > neeq2i | Structured version Visualization version GIF version | ||
| Description: Inference for inequality. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) |
| Ref | Expression |
|---|---|
| neeq1i.1 | ⊢ 𝐴 = 𝐵 |
| Ref | Expression |
|---|---|
| neeq2i | ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | neeq1i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
| 2 | 1 | eqeq2i 2750 | . 2 ⊢ (𝐶 = 𝐴 ↔ 𝐶 = 𝐵) |
| 3 | 2 | necon3bii 2993 | 1 ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ≠ wne 2940 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ne 2941 |
| This theorem is referenced by: neeqtri 3013 omsucne 7906 suppvalbr 8189 nosgnn0 27703 upgr3v3e3cycl 30199 upgr4cycl4dv4e 30204 disjdsct 32712 divnumden2 32817 usgrgt2cycl 35135 onov0suclim 43287 |
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