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Mirrors > Home > MPE Home > Th. List > neqcomd | Structured version Visualization version 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 2740 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
3 | 1, 2 | sylnib 328 | 1 ⊢ (𝜑 → ¬ 𝐵 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2725 |
This theorem is referenced by: phpeqd 9165 phpeqdOLD 9175 simpgnsgd 19887 aks4d1p8d2 40592 rr-phpd 42575 |
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