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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfext | Structured version Visualization version GIF version |
Description: Closed form of nfex 2334. (Contributed by BJ, 10-Oct-2019.) |
Ref | Expression |
---|---|
bj-nfext | ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nf5 2281 | . . . . 5 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑)) | |
2 | 1 | biimpi 217 | . . . 4 ⊢ (Ⅎ𝑦𝜑 → ∀𝑦(𝜑 → ∀𝑦𝜑)) |
3 | 2 | alimi 1803 | . . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑)) |
4 | nfa2 2166 | . . . 4 ⊢ Ⅎ𝑦∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) | |
5 | bj-hbext 33941 | . . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)) | |
6 | 4, 5 | alrimi 2203 | . . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)) |
7 | 3, 6 | syl 17 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)) |
8 | nf5 2281 | . 2 ⊢ (Ⅎ𝑦∃𝑥𝜑 ↔ ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)) | |
9 | 7, 8 | sylibr 235 | 1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 ∃wex 1771 Ⅎwnf 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-or 842 df-ex 1772 df-nf 1776 |
This theorem is referenced by: (None) |
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