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| Mirrors > Home > MPE Home > Th. List > nfrd | Structured version Visualization version GIF version | ||
| Description: Consequence of the definition of not-free in a context. (Contributed by Wolf Lammen, 15-Oct-2021.) | 
| Ref | Expression | 
|---|---|
| nfrd.1 | ⊢ (𝜑 → Ⅎ𝑥𝜓) | 
| Ref | Expression | 
|---|---|
| nfrd | ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfrd.1 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 2 | df-nf 1783 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓)) | |
| 3 | 1, 2 | sylib 218 | 1 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 ∃wex 1778 Ⅎwnf 1782 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 | 
| This theorem depends on definitions: df-bi 207 df-nf 1783 | 
| This theorem is referenced by: 19.38a 1839 19.38b 1840 nfimd 1893 nf5r 2193 19.9d 2202 nfald 2327 exists2 2661 eusv2i 5393 bj-nfimt 36640 bj-nfald 37139 eu6w 42691 | 
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