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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 1858 changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
Ref | Expression |
---|---|
nf5r | ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 2206 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
2 | df-nf 1858 | . . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
3 | 2 | biimpi 206 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑)) |
4 | 1, 3 | syl5 34 | 1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1629 ∃wex 1852 Ⅎwnf 1856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-12 2203 |
This theorem depends on definitions: df-bi 197 df-ex 1853 df-nf 1858 |
This theorem is referenced by: nf5ri 2219 nf5rd 2220 19.21tOLDOLD 2230 sbft 2526 bj-alrim 33020 bj-nexdt 33024 bj-cbv3tb 33048 bj-nfs1t2 33052 bj-sbftv 33099 bj-equsal1t 33144 stdpc5t 33149 bj-axc14 33173 wl-nfeqfb 33658 |
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