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Theorem 19.3v 1896
Description: Version of 19.3 2068 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 1895. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 1934. (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 1752 . 2 (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
2 19.9v 1895 . . 3 (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑)
32con2bii 347 . 2 (𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
41, 3bitr4i 267 1 (∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1480  wex 1703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887
This theorem depends on definitions:  df-bi 197  df-ex 1704
This theorem is referenced by:  spvw  1897  19.27v  1907  19.28v  1908  19.37v  1909  axrep1  4770  kmlem14  8982  zfcndrep  9433  zfcndpow  9435  zfcndac  9438  bj-axrep1  32772  bj-snsetex  32935  iooelexlt  33190  dford4  37422  relexp0eq  37819
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