Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  19.9alt Structured version   Visualization version   GIF version

Theorem 19.9alt 33270
 Description: Version of 19.9t 2059 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 1767 . . . 4 (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
2 19.9t 2059 . . . 4 (Ⅎ𝑥 ¬ 𝜑 → (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑))
31, 2syl 17 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑))
43con2bid 343 . 2 (Ⅎ𝑥𝜑 → (𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑))
5 alex 1743 . 2 (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
64, 5syl6rbbr 278 1 (Ⅎ𝑥𝜑 → (∀𝑥𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator