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Theorem bj-nfext 34046
Description: Closed form of nfex 2339. (Contributed by BJ, 10-Oct-2019.)
Assertion
Ref Expression
bj-nfext (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)

Proof of Theorem bj-nfext
StepHypRef Expression
1 nf5 2286 . . . . 5 (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
21biimpi 218 . . . 4 (Ⅎ𝑦𝜑 → ∀𝑦(𝜑 → ∀𝑦𝜑))
32alimi 1808 . . 3 (∀𝑥𝑦𝜑 → ∀𝑥𝑦(𝜑 → ∀𝑦𝜑))
4 nfa2 2172 . . . 4 𝑦𝑥𝑦(𝜑 → ∀𝑦𝜑)
5 bj-hbext 34044 . . . 4 (∀𝑥𝑦(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦𝑥𝜑))
64, 5alrimi 2209 . . 3 (∀𝑥𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∃𝑥𝜑 → ∀𝑦𝑥𝜑))
73, 6syl 17 . 2 (∀𝑥𝑦𝜑 → ∀𝑦(∃𝑥𝜑 → ∀𝑦𝑥𝜑))
8 nf5 2286 . 2 (Ⅎ𝑦𝑥𝜑 ↔ ∀𝑦(∃𝑥𝜑 → ∀𝑦𝑥𝜑))
97, 8sylibr 236 1 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  wex 1776  wnf 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2157  ax-12 2173
This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1777  df-nf 1781
This theorem is referenced by: (None)
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