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| Mirrors > Home > MPE Home > Th. List > nfcrii | Structured version Visualization version GIF version | ||
| Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2140, ax-11 2156. (Revised by GG, 23-May-2024.) | 
| Ref | Expression | 
|---|---|
| nfcrii.1 | ⊢ Ⅎ𝑥𝐴 | 
| Ref | Expression | 
|---|---|
| nfcrii | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfcrii.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 2 | 1 | nfcri 2896 | . 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 | 
| 3 | 2 | nf5ri 2194 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 ∈ wcel 2107 Ⅎwnfc 2889 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-nf 1783 df-clel 2815 df-nfc 2891 | 
| This theorem is referenced by: bnj1230 34817 bnj1000 34956 bnj1204 35027 bnj1307 35038 bnj1311 35039 bnj1398 35049 bnj1466 35068 bnj1467 35069 bnj1523 35086 | 
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