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Mirrors > Home > ILE Home > Th. List > nesym | GIF version |
Description: Characterization of inequality in terms of reversed equality (see bicom 140). (Contributed by BJ, 7-Jul-2018.) |
Ref | Expression |
---|---|
nesym | ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2179 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
2 | 1 | necon3abii 2383 | 1 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 105 = wceq 1353 ≠ wne 2347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-5 1447 ax-gen 1449 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-cleq 2170 df-ne 2348 |
This theorem is referenced by: nesymi 2393 nesymir 2394 0neqopab 5916 fzdifsuc 10075 isprm3 12109 |
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