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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 1701 changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
Ref | Expression |
---|---|
nf5r | ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 2039 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
2 | df-nf 1701 | . . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
3 | 2 | biimpi 205 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑)) |
4 | 1, 3 | syl5 33 | 1 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 ∃wex 1695 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-nf 1701 |
This theorem is referenced by: nf5ri 2053 nf5rd 2054 19.21t 2061 sbft 2367 bj-alrim 31870 bj-nexdt 31874 bj-cbv3tb 31898 bj-nfs1t2 31902 bj-sbftv 31951 bj-equsal1t 31997 stdpc5t 32002 bj-axc14 32032 wl-nfeqfb 32502 |
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