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Mirrors > Home > ILE Home > Th. List > nesym | GIF version |
Description: Characterization of inequality in terms of reversed equality (see bicom 139). (Contributed by BJ, 7-Jul-2018.) |
Ref | Expression |
---|---|
nesym | ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2119 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
2 | 1 | necon3abii 2321 | 1 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 = wceq 1316 ≠ wne 2285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-5 1408 ax-gen 1410 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-cleq 2110 df-ne 2286 |
This theorem is referenced by: nesymi 2331 nesymir 2332 0neqopab 5784 fzdifsuc 9829 isprm3 11726 |
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