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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 1780 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 2178 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
2 | id 22 | . . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑) | |
3 | 2 | nfrd 1787 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑)) |
4 | 1, 3 | syl5 34 | 1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 ∃wex 1775 Ⅎwnf 1779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-12 2174 |
This theorem depends on definitions: df-bi 207 df-ex 1776 df-nf 1780 |
This theorem is referenced by: nf5rd 2193 19.3t 2198 sbft 2267 bj-alrim 36675 bj-nexdt 36679 bj-cbv3tb 36769 bj-nfs1t2 36773 bj-equsal1t 36804 stdpc5t 36809 bj-axc14 36838 wl-nfeqfb 37516 |
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