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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 2942 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | nfned.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfned.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 2, 3 | nfeqd 2915 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵) |
5 | 4 | nfnd 1866 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 = 𝐵) |
6 | 1, 5 | nfxfrd 1861 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 Ⅎwnf 1791 Ⅎwnfc 2885 ≠ wne 2941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-cleq 2730 df-nfc 2887 df-ne 2942 |
This theorem is referenced by: (None) |
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