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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfext | Structured version Visualization version GIF version |
Description: Closed form of nfex 2318. (Contributed by BJ, 10-Oct-2019.) |
Ref | Expression |
---|---|
bj-nfext | ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nf5 2279 | . . . . 5 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑)) | |
2 | 1 | biimpi 215 | . . . 4 ⊢ (Ⅎ𝑦𝜑 → ∀𝑦(𝜑 → ∀𝑦𝜑)) |
3 | 2 | alimi 1814 | . . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑)) |
4 | nfa2 2170 | . . . 4 ⊢ Ⅎ𝑦∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) | |
5 | bj-hbext 34900 | . . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)) | |
6 | 4, 5 | alrimi 2206 | . . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)) |
7 | 3, 6 | syl 17 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)) |
8 | nf5 2279 | . 2 ⊢ (Ⅎ𝑦∃𝑥𝜑 ↔ ∀𝑦(∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)) | |
9 | 7, 8 | sylibr 233 | 1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 ∃wex 1782 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-or 845 df-ex 1783 df-nf 1787 |
This theorem is referenced by: (None) |
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