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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfalt | Structured version Visualization version GIF version |
Description: Closed form of nfal 2317. (Contributed by BJ, 2-May-2019.) |
Ref | Expression |
---|---|
bj-nfalt | ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-hbalt 35175 | . . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) | |
2 | 1 | alimi 1814 | . . 3 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) |
3 | 2 | alcoms 2156 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) |
4 | nf5 2279 | . . 3 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑)) | |
5 | 4 | albii 1822 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑)) |
6 | nf5 2279 | . 2 ⊢ (Ⅎ𝑦∀𝑥𝜑 ↔ ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) | |
7 | 3, 5, 6 | 3imtr4i 292 | 1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 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 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-or 847 df-ex 1783 df-nf 1787 |
This theorem is referenced by: bj-dvelimdv1 35347 |
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