Theorem 19.42v 1962

Description: Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
19.42v	|- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )

Distinct variable group:   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem 19.42v

Step	Hyp	Ref	Expression
1		ax-17 1579	. 2 |- ( ph -> A. x ph )
2	1	19.42h 1739	1 |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   E.wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This proof depends on definitions:  df-bi 117

This theorem is used by:  exdistr  1965  19.42vv  1967  19.42vvv  1968  4exdistr  1972  cbvex2  1978  2sb5  2043  2sb5rf  2049  rexcom4a  2846  ceqsex2  2863  reuind  3031  2rmorex  3032  sbccomlem  3126  bm1.3ii  4254  opm  4374  eqvinop  4383  uniuni  4597  elco  4946  dmopabss  4993  dmopab3  4994  mptpreima  5281  brprcneu  5688  relelfvdm  5727  fndmin  5816  fliftf  6005  dfoprab2  6135  dmoprab  6169  dmoprabss  6170  fnoprabg  6189  opabex3d  6350  opabex3  6351  eroveu  6900  dmaddpq  7746  dmmulpq  7747  prarloc  7870  ltexprlemopl  7968  ltexprlemlol  7969  ltexprlemopu  7970  ltexprlemupu  7971  shftdm  11587  fngzsum  13708  gzsumvalx  13709  ntreq0  15233  bdbm1.3ii  16917