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Mirrors > Home > ILE Home > Th. List > nesym | GIF version |
Description: Characterization of inequality in terms of reversed equality (see bicom 138). (Contributed by BJ, 7-Jul-2018.) |
Ref | Expression |
---|---|
nesym | ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2083 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
2 | 1 | necon3abii 2281 | 1 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 103 = wceq 1284 ≠ wne 2245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-5 1376 ax-gen 1378 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-cleq 2074 df-ne 2246 |
This theorem is referenced by: nesymi 2291 nesymir 2292 0neqopab 5570 fzdifsuc 9098 isprm3 10500 |
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