|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 3047 | . 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵) | |
| 2 | nfneld.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 3 | nfneld.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
| 4 | 2, 3 | nfeld 2917 | . . 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 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ∉ wnel 3046 | 
| 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-8 2110 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-clel 2816 df-nfc 2892 df-nel 3047 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |