Theorem 19.3v 1897

Description: Version of 19.3 2069 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 1896. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 1935. (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

Step	Hyp	Ref	Expression
1		alex 1753	. 2 |- ( A. x ph <-> -. E. x -. ph )
2		19.9v 1896	. . 3 |- ( E. x -. ph <-> -. ph )
3	2	con2bii 347	. 2 |- ( ph <-> -. E. x -. ph )
4	1, 3	bitr4i 267	1 |- ( A. x ph <-> ph )

Syntax hints:   -. wn 3   <-> wb 196   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  spvw  1898  19.27v  1908  19.28v  1909  19.37v  1910  axrep1  4772  kmlem14  8985  zfcndrep  9436  zfcndpow  9438  zfcndac  9441  bj-axrep1  32788  bj-snsetex  32951  iooelexlt  33210  dford4  37596  relexp0eq  37993