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Mirrors > Home > ILE Home > Th. List > nfalt | GIF version |
Description: Closed form of nfal 1569. (Contributed by Jim Kingdon, 11-May-2018.) |
Ref | Expression |
---|---|
nfalt | ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥∀𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alim 1450 | . . . 4 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)) | |
2 | alcom 1471 | . . . 4 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑) | |
3 | 1, 2 | syl6ib 160 | . . 3 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)) |
4 | 3 | alimi 1448 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)) |
5 | df-nf 1454 | . . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
6 | 5 | albii 1463 | . . 3 ⊢ (∀𝑦Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑)) |
7 | alcom 1471 | . . 3 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)) | |
8 | 6, 7 | bitri 183 | . 2 ⊢ (∀𝑦Ⅎ𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑)) |
9 | df-nf 1454 | . 2 ⊢ (Ⅎ𝑥∀𝑦𝜑 ↔ ∀𝑥(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)) | |
10 | 4, 8, 9 | 3imtr4i 200 | 1 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥∀𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1346 Ⅎwnf 1453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-7 1441 ax-gen 1442 |
This theorem depends on definitions: df-bi 116 df-nf 1454 |
This theorem is referenced by: dvelimor 2011 |
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