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Mirrors > Home > MPE Home > Th. List > nfneld | Structured version Visualization version 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 2898 | . 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵) | |
2 | nfneld.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfneld.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 2, 3 | nfeld 2773 | . . 3 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵) |
5 | 4 | nfnd 1785 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝐴 ∈ 𝐵) |
6 | 1, 5 | nfxfrd 1780 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∉ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 ∉ wnel 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 df-cleq 2615 df-clel 2618 df-nfc 2753 df-nel 2898 |
This theorem is referenced by: (None) |
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