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Theorem 19.9alt 34071
Description: Version of 19.9t 2069 for universal quantifier. (Contributed by NM, 9-Nov-2020.)
Assertion
Ref Expression
19.9alt (Ⅎ𝑥𝜑 → (∀𝑥𝜑𝜑))

Proof of Theorem 19.9alt
StepHypRef Expression
1 nfnt 1780 . . . 4 (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
2 19.9t 2069 . . . 4 (Ⅎ𝑥 ¬ 𝜑 → (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑))
31, 2syl 17 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑))
43con2bid 344 . 2 (Ⅎ𝑥𝜑 → (𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑))
5 alex 1751 . 2 (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
64, 5syl6rbbr 279 1 (Ⅎ𝑥𝜑 → (∀𝑥𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1479  wex 1702  wnf 1706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1703  df-nf 1708
This theorem is referenced by: (None)
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