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Theorem nf5-1 2134
Description: One direction of nf5 2272 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021.)
Assertion
Ref Expression
nf5-1 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

Proof of Theorem nf5-1
StepHypRef Expression
1 exim 1829 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥𝑥𝜑))
2 hbe1a 2133 . . 3 (∃𝑥𝑥𝜑 → ∀𝑥𝜑)
31, 2syl6 35 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))
43nfd 1785 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1532  wex 1774  wnf 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-10 2130
This theorem depends on definitions:  df-bi 206  df-ex 1775  df-nf 1779
This theorem is referenced by:  nf5i  2135  nf5dh  2136  nf5d  2274  hbnt  2284  19.9ht  2309
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