|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > nfcrd | Structured version Visualization version GIF version | ||
| Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) | 
| Ref | Expression | 
|---|---|
| nfcrd.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) | 
| Ref | Expression | 
|---|---|
| nfcrd | ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfcrd.1 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 2 | nfcr 2894 | . 2 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 Ⅎwnf 1782 ∈ 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 | 
| 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: nfeld 2916 dvelimdc 2929 nfraldw 3308 nfcsbd 3923 nfcsbw 3924 nfifd 4554 axextnd 10632 axrepndlem1 10633 axunndlem1 10636 axregnd 10645 axsepg2 35097 axsepg2ALT 35098 axextdist 35801 nfintd 49247 nfiund 49248 nfiundg 49249 | 
| Copyright terms: Public domain | W3C validator |