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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 2746 | . 2 ⊢ (𝐶 = 𝐴 ↔ 𝐶 = 𝐵) |
3 | 2 | necon3bii 2993 | 1 ⊢ (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ≠ wne 2940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2725 df-ne 2941 |
This theorem is referenced by: neeqtri 3013 omsucne 7822 suppvalbr 8097 nosgnn0 27022 upgr3v3e3cycl 29166 upgr4cycl4dv4e 29171 disjdsct 31663 divnumden2 31763 usgrgt2cycl 33781 onov0suclim 41652 |
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