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| Mirrors > Home > MPE Home > Th. List > neqned | Structured version Visualization version GIF version | ||
| Description: If it is not the case that two classes are equal, then they are unequal. Converse of neneqd 2945. One-way deduction form of df-ne 2941. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2967. (Revised by Wolf Lammen, 22-Nov-2019.) |
| Ref | Expression |
|---|---|
| neqned.1 | ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| neqned | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | neqned.1 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
| 2 | df-ne 2941 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
| 3 | 1, 2 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
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