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Mirrors > Home > ILE Home > Th. List > neqcomd | GIF version |
Description: Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
neqcomd.1 | ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
Ref | Expression |
---|---|
neqcomd | ⊢ (𝜑 → ¬ 𝐵 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neqcomd.1 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
2 | eqcom 2189 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
3 | 1, 2 | sylnib 677 | 1 ⊢ (𝜑 → ¬ 𝐵 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1363 |
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 615 ax-in2 616 ax-5 1457 ax-gen 1459 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-cleq 2180 |
This theorem is referenced by: logbgcd1irraplemexp 14613 |
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