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Theorem bj-nfext 34902
Description: Closed form of nfex 2318. (Contributed by BJ, 10-Oct-2019.)
Assertion
Ref Expression
bj-nfext (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)

Proof of Theorem bj-nfext
StepHypRef Expression
1 nf5 2279 . . . . 5 (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
21biimpi 215 . . . 4 (Ⅎ𝑦𝜑 → ∀𝑦(𝜑 → ∀𝑦𝜑))
32alimi 1814 . . 3 (∀𝑥𝑦𝜑 → ∀𝑥𝑦(𝜑 → ∀𝑦𝜑))
4 nfa2 2170 . . . 4 𝑦𝑥𝑦(𝜑 → ∀𝑦𝜑)
5 bj-hbext 34900 . . . 4 (∀𝑥𝑦(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦𝑥𝜑))
64, 5alrimi 2206 . . 3 (∀𝑥𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∃𝑥𝜑 → ∀𝑦𝑥𝜑))
73, 6syl 17 . 2 (∀𝑥𝑦𝜑 → ∀𝑦(∃𝑥𝜑 → ∀𝑦𝑥𝜑))
8 nf5 2279 . 2 (Ⅎ𝑦𝑥𝜑 ↔ ∀𝑦(∃𝑥𝜑 → ∀𝑦𝑥𝜑))
97, 8sylibr 233 1 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1537  wex 1782  wnf 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783  df-nf 1787
This theorem is referenced by: (None)
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