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Theorem nf5-1 2177
Description: One direction of nf5 2278 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 1908 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥𝑥𝜑))
2 hbe1a 2176 . . 3 (∃𝑥𝑥𝜑 → ∀𝑥𝜑)
31, 2syl6 35 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))
43nfd 1863 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1628  wex 1851  wnf 1855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-10 2173
This theorem depends on definitions:  df-bi 197  df-ex 1852  df-nf 1857
This theorem is referenced by:  nf5i  2178  nf5dh  2179  nf5dvOLD  2181  nf5d  2280  hbnt  2308
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