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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 1779 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 2170 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
2 | id 22 | . . 3 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑) | |
3 | 2 | nfrd 1786 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑)) |
4 | 1, 3 | syl5 34 | 1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1532 ∃wex 1774 Ⅎwnf 1778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-12 2167 |
This theorem depends on definitions: df-bi 206 df-ex 1775 df-nf 1779 |
This theorem is referenced by: nf5rd 2185 19.3t 2190 sbft 2257 bj-alrim 36165 bj-nexdt 36169 bj-cbv3tb 36259 bj-nfs1t2 36263 bj-equsal1t 36294 stdpc5t 36299 bj-axc14 36328 wl-nfeqfb 36998 |
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