Theorem 19.36v 1904

Description: Version of 19.36 2098 with a dv condition instead of a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)

Assertion

Ref	Expression
19.36v	|- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )

Distinct variable group:   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem 19.36v

Step	Hyp	Ref	Expression
1		19.35 1805	. 2 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) )
2		19.9v 1896	. . 3 |- ( E. x ps <-> ps )
3	2	imbi2i 326	. 2 |- ( ( A. x ph -> E. x ps ) <-> ( A. x ph -> ps ) )
4	1, 3	bitri 264	1 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   <-> 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:  19.36iv  1905  19.12vvv  1907  19.12vv  2180  ax13lem2  2296  axext2  2603  vtocl2  3261  vtocl3  3262  bnj1090  31047  bj-spimvwt  32656  bj-spcimdv  32884  bj-spcimdvv  32885  19.36vv  38582