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Theorem 19.3v 1884
 Description: Version of 19.3 2057 with a dv condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 1883. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 1922. (Revised by Wolf Lammen, 4-Dec-2017.)
Assertion
Ref Expression
19.3v (∀𝑥𝜑𝜑)
Distinct variable group:   𝜑,𝑥

Proof of Theorem 19.3v
StepHypRef Expression
1 alex 1743 . 2 (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
2 19.9v 1883 . . 3 (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑)
32con2bii 346 . 2 (𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
41, 3bitr4i 266 1 (∀𝑥𝜑𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195  ∀wal 1473  ∃wex 1695 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 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by:  spvw  1885  19.27v  1895  19.28v  1896  19.37v  1897  axrep1  4700  kmlem14  8868  zfcndrep  9315  zfcndpow  9317  zfcndac  9320  bj-axrep1  31976  bj-snsetex  32144  iooelexlt  32386  dford4  36614  relexp0eq  37012
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