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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 2827 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
3 | 1, 2 | sylnib 330 | 1 ⊢ (𝜑 → ¬ 𝐵 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 |
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 1969 ax-7 2014 ax-9 2123 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-cleq 2813 |
This theorem is referenced by: phpeqd 8699 simpgnsgd 19215 rr-phpd 40637 |
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