| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 2744 | . 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 1540 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 |
| This theorem is referenced by: phpeqd 9252 phpeqdOLD 9262 simpgnsgd 20120 aks4d1p8d2 42086 selvvvval 42595 rr-phpd 44222 |
| Copyright terms: Public domain | W3C validator |