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Mirrors > Home > ILE Home > Th. List > nesym | GIF version |
Description: Characterization of inequality in terms of reversed equality (see bicom 128). (Contributed by BJ, 7-Jul-2018.) |
Ref | Expression |
---|---|
nesym | ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2042 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
2 | 1 | necon3abii 2241 | 1 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 98 = wceq 1243 ≠ wne 2204 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-5 1336 ax-gen 1338 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 df-ne 2206 |
This theorem is referenced by: nesymi 2251 nesymir 2252 0neqopab 5550 fzdifsuc 8943 |
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