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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 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
Ref | Expression |
---|---|
nf5r | ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 2052 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
2 | df-nf 1710 | . . 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 1481 ∃wex 1704 Ⅎwnf 1708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-ex 1705 df-nf 1710 |
This theorem is referenced by: nf5ri 2065 nf5rd 2066 19.21tOLDOLD 2074 sbft 2379 bj-alrim 32683 bj-nexdt 32687 bj-cbv3tb 32711 bj-nfs1t2 32715 bj-sbftv 32763 bj-equsal1t 32809 stdpc5t 32814 bj-axc14 32839 wl-nfeqfb 33323 |
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