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| Mirrors > Home > MPE Home > Th. List > nfned | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.) |
| Ref | Expression |
|---|---|
| nfned.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
| nfned.2 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
| Ref | Expression |
|---|---|
| nfned | ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ne 2941 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
| 2 | nfned.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 3 | nfned.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
| 4 | 2, 3 | nfeqd 2916 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵) |
| 5 | 4 | nfnd 1858 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 = 𝐵) |
| 6 | 1, 5 | nfxfrd 1854 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 Ⅎwnf 1783 Ⅎwnfc 2890 ≠ wne 2940 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-cleq 2729 df-nfc 2892 df-ne 2941 |
| This theorem is referenced by: (None) |
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