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Theorem bj-nfalt 32371
Description: Closed form of nfal 2150. (Contributed by BJ, 2-May-2019.)
Assertion
Ref Expression
bj-nfalt (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)

Proof of Theorem bj-nfalt
StepHypRef Expression
1 bj-hbalt 32340 . . . 4 (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝑥𝜑))
21alimi 1736 . . 3 (∀𝑦𝑥(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦𝑥𝜑))
32alcoms 2032 . 2 (∀𝑥𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦𝑥𝜑))
4 nf5 2113 . . 3 (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
54albii 1744 . 2 (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦(𝜑 → ∀𝑦𝜑))
6 nf5 2113 . 2 (Ⅎ𝑦𝑥𝜑 ↔ ∀𝑦(∀𝑥𝜑 → ∀𝑦𝑥𝜑))
73, 5, 63imtr4i 281 1 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1702  df-nf 1707
This theorem is referenced by:  bj-dvelimdv1  32507
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