Theorem 19.37v 1834

Description: Version of 19.37 2065 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)

Assertion

Ref	Expression
19.37v	|- ( E. x ( ph -> ps ) <-> ( ph -> E. x ps ) )

Distinct variable group:   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem 19.37v

Step	Hyp	Ref	Expression
1		19.35 1748	. 2 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) )
2		19.3v 1821	. . 3 |- ( A. x ph <-> ph )
3	2	imbi1i 332	. 2 |- ( ( A. x ph -> E. x ps ) <-> ( ph -> E. x ps ) )
4	1, 3	bitri 257	1 |- ( E. x ( ph -> ps ) <-> ( ph -> E. x ps ) )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1450   E.wex 1671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813

This theorem depends on definitions:  df-bi 190  df-ex 1672

This theorem is referenced by:  19.37iv  1835  axrep5  4513  fvn0ssdmfun  6028  kmlem14  8611  kmlem15  8612  eqvincg  28189  bnj132  29604  bnj1098  29667  bnj150  29759  bnj865  29806  bnj996  29838  bnj1021  29847  bnj1090  29860  bnj1176  29886  bj-axrep5  31473  cnvssco  36283  refimssco  36284  19.37vv  36804  pm11.61  36813  relopabVD  37361  rmoanim  38745