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| Mirrors > Home > MPE Home > Th. List > nf5r | Structured version Visualization version GIF version | ||
| Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1784 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 23-Nov-2023.) |
| Ref | Expression |
|---|---|
| nf5r | ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 19.8a 2181 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
| 2 | id 22 | . . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑) | |
| 3 | 2 | nfrd 1791 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑)) |
| 4 | 1, 3 | syl5 34 | 1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 ∃wex 1779 Ⅎwnf 1783 |
| 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-12 2177 |
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: nf5rd 2196 19.3t 2201 sbft 2270 bj-alrim 36694 bj-nexdt 36698 bj-cbv3tb 36788 bj-nfs1t2 36792 bj-equsal1t 36823 stdpc5t 36828 bj-axc14 36857 wl-nfeqfb 37537 |
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