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Theorem 19.3v 1763
Description: Version of 19.3 1896 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 1762. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 1798. (Revised by Wolf Lammen, 4-Dec-2017.)
Assertion
Ref Expression
19.3v  |-  ( A. x ph  <->  ph )
Distinct variable group:    ph, x

Proof of Theorem 19.3v
StepHypRef Expression
1 alex 1655 . 2  |-  ( A. x ph  <->  -.  E. x  -.  ph )
2 19.9v 1762 . . 3  |-  ( E. x  -.  ph  <->  -.  ph )
32con2bii 330 . 2  |-  ( ph  <->  -. 
E. x  -.  ph )
41, 3bitr4i 252 1  |-  ( A. x ph  <->  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184   A.wal 1397   E.wex 1620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755
This theorem depends on definitions:  df-bi 185  df-ex 1621
This theorem is referenced by:  spvw  1764  19.27v  1774  19.28v  1775  19.37v  1776  axrep1  4479  kmlem14  8456  zfcndrep  8903  zfcndpow  8905  zfcndac  8908  dford4  31137  bj-axrep1  34721  bj-snsetex  34869  relexp0eq  38236
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