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Theorem bj-nfext 32366
Description: Closed form of nfex 2151. (Contributed by BJ, 10-Oct-2019.)
Assertion
Ref Expression
bj-nfext (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)

Proof of Theorem bj-nfext
StepHypRef Expression
1 nf5 2113 . . . . 5 (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
21biimpi 206 . . . 4 (Ⅎ𝑦𝜑 → ∀𝑦(𝜑 → ∀𝑦𝜑))
32alimi 1736 . . 3 (∀𝑥𝑦𝜑 → ∀𝑥𝑦(𝜑 → ∀𝑦𝜑))
4 nfa2 2037 . . . 4 𝑦𝑥𝑦(𝜑 → ∀𝑦𝜑)
5 bj-hbext 32364 . . . 4 (∀𝑥𝑦(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦𝑥𝜑))
64, 5alrimi 2080 . . 3 (∀𝑥𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∃𝑥𝜑 → ∀𝑦𝑥𝜑))
73, 6syl 17 . 2 (∀𝑥𝑦𝜑 → ∀𝑦(∃𝑥𝜑 → ∀𝑦𝑥𝜑))
8 nf5 2113 . 2 (Ⅎ𝑦𝑥𝜑 ↔ ∀𝑦(∃𝑥𝜑 → ∀𝑦𝑥𝜑))
97, 8sylibr 224 1 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wex 1701  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: (None)
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