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| Mirrors > Home > MPE Home > Th. List > nf5rd | Structured version Visualization version GIF version | ||
| Description: Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) | 
| Ref | Expression | 
|---|---|
| nf5rd.1 | ⊢ (𝜑 → Ⅎ𝑥𝜓) | 
| Ref | Expression | 
|---|---|
| nf5rd | ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nf5rd.1 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 2 | nf5r 2193 | . 2 ⊢ (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓)) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 Ⅎwnf 1782 | 
| 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-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-ex 1779 df-nf 1783 | 
| This theorem is referenced by: spimedv 2196 alrimdd 2213 nf5di 2284 hbnt 2293 hbimd 2297 dvelimhw 2346 dveeq2 2382 dveeq1 2384 axc9 2386 spimed 2392 dvelimh 2454 abidnf 3707 eusvnfb 5392 axrepnd 10635 axacndlem4 10651 bj-cbv2v 36800 bj-elgab 36941 wl-nfeqfb 37538 | 
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