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Theorem bj-nfalt 13298
 Description: Closed form of nfal 1556 (copied from set.mm). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nfalt (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)

Proof of Theorem bj-nfalt
StepHypRef Expression
1 bj-hbalt 13297 . . . 4 (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝑥𝜑))
21alimi 1435 . . 3 (∀𝑦𝑥(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦𝑥𝜑))
32alcoms 1456 . 2 (∀𝑥𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦𝑥𝜑))
4 df-nf 1441 . . 3 (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
54albii 1450 . 2 (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦(𝜑 → ∀𝑦𝜑))
6 df-nf 1441 . 2 (Ⅎ𝑦𝑥𝜑 ↔ ∀𝑦(∀𝑥𝜑 → ∀𝑦𝑥𝜑))
73, 5, 63imtr4i 200 1 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1333  Ⅎwnf 1440 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 1427  ax-7 1428  ax-gen 1429 This theorem depends on definitions:  df-bi 116  df-nf 1441 This theorem is referenced by: (None)
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