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Theorem 19.3v 1720
Description: Special case of Theorem 19.3 of [Margaris] p. 89 (see 19.3 1827). Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 1719. (Contributed by Anthony Hart, 13-Sep-2011.) (Revised by NM, 1-Aug-2017.) Remove a dependency on ax-7 1730. (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 1618 . 2  |-  ( A. x ph  <->  -.  E. x  -.  ph )
2 19.9v 1719 . . 3  |-  ( E. x  -.  ph  <->  -.  ph )
32con2bii 332 . 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 1368   E.wex 1587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710
This theorem depends on definitions:  df-bi 185  df-ex 1588
This theorem is referenced by:  spvw  1721  axrep1  4513  kmlem14  8444  dford4  29527  bj-axrep1  32642  bj-snsetex  32789
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