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Theorem bj-nfalt 10726
Description: Closed form of nfal 1509 (copied from set.mm). (Contributed by BJ, 2-May-2019.)
Assertion
Ref Expression
bj-nfalt (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)

Proof of Theorem bj-nfalt
StepHypRef Expression
1 df-nf 1391 . . . 4 (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
21albii 1400 . . 3 (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦(𝜑 → ∀𝑦𝜑))
3 bj-hbalt 10725 . . . . 5 (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝑥𝜑))
43alimi 1385 . . . 4 (∀𝑦𝑥(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦𝑥𝜑))
54alcoms 1406 . . 3 (∀𝑥𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦𝑥𝜑))
62, 5sylbi 119 . 2 (∀𝑥𝑦𝜑 → ∀𝑦(∀𝑥𝜑 → ∀𝑦𝑥𝜑))
7 df-nf 1391 . 2 (Ⅎ𝑦𝑥𝜑 ↔ ∀𝑦(∀𝑥𝜑 → ∀𝑦𝑥𝜑))
86, 7sylibr 132 1 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wnf 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by: (None)
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