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Theorem 19.3v 1954
Description: Version of 19.3 2107 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 1953. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 1981. (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 1793 . 2 (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
2 19.9v 1953 . . 3 (∃𝑥 ¬ 𝜑 ↔ ¬ 𝜑)
32con2bii 346 . 2 (𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
41, 3bitr4i 267 1 (∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945
This theorem depends on definitions:  df-bi 197  df-ex 1745
This theorem is referenced by:  spvw  1955  19.27v  1964  19.28v  1965  19.37v  1966  axrep1  4805  kmlem14  9023  zfcndrep  9474  zfcndpow  9476  zfcndac  9479  bj-axrep1  32913  bj-snsetex  33076  iooelexlt  33340  dford4  37913  relexp0eq  38310
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