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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 2937 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | nfned.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfned.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 2, 3 | nfeqd 2909 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵) |
5 | 4 | nfnd 1854 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 = 𝐵) |
6 | 1, 5 | nfxfrd 1849 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 Ⅎwnf 1778 Ⅎwnfc 2879 ≠ wne 2936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-nf 1779 df-cleq 2720 df-nfc 2881 df-ne 2937 |
This theorem is referenced by: (None) |
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