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Mirrors > Home > ILE Home > Th. List > nfneld | GIF version |
Description: Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Ref | Expression |
---|---|
nfneld.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfneld.2 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfneld | ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∉ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 2402 | . 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵) | |
2 | nfneld.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfneld.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 2, 3 | nfeld 2295 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵) |
5 | 4 | nfnd 1635 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 ∈ 𝐵) |
6 | 1, 5 | nfxfrd 1451 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∉ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 Ⅎwnf 1436 ∈ wcel 1480 Ⅎwnfc 2266 ∉ wnel 2401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-cleq 2130 df-clel 2133 df-nfc 2268 df-nel 2402 |
This theorem is referenced by: (None) |
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