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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nfalt | GIF version |
Description: Closed form of nfal 1468 (copied from set.mm). (Contributed by BJ, 2-May-2019.) |
Ref | Expression |
---|---|
bj-nfalt | ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nf 1350 | . . . 4 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑)) | |
2 | 1 | albii 1359 | . . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑)) |
3 | bj-hbalt 9903 | . . . . 5 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) | |
4 | 3 | alimi 1344 | . . . 4 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) |
5 | 4 | alcoms 1365 | . . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) |
6 | 2, 5 | sylbi 114 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) |
7 | df-nf 1350 | . 2 ⊢ (Ⅎ𝑦∀𝑥𝜑 ↔ ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) | |
8 | 6, 7 | sylibr 137 | 1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 Ⅎwnf 1349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: (None) |
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