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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfalt | Structured version Visualization version GIF version |
Description: Closed form of nfal 2341. (Contributed by BJ, 2-May-2019.) |
Ref | Expression |
---|---|
bj-nfalt | ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-hbalt 34019 | . . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) | |
2 | 1 | alimi 1811 | . . 3 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) |
3 | 2 | alcoms 2161 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) |
4 | nf5 2289 | . . 3 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑)) | |
5 | 4 | albii 1819 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑)) |
6 | nf5 2289 | . 2 ⊢ (Ⅎ𝑦∀𝑥𝜑 ↔ ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) | |
7 | 3, 5, 6 | 3imtr4i 294 | 1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 Ⅎwnf 1783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-11 2160 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1780 df-nf 1784 |
This theorem is referenced by: bj-dvelimdv1 34180 |
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