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  4256  opm  4376  eqvinop  4385  uniuni  4599  elco  4948  dmopabss  4995  dmopab3  4996  mptpreima  5284  brprcneu  5691  relelfvdm  5730  fndmin  5819  fliftf  6008  dfoprab2  6138  dmoprab  6172  dmoprabss  6173  fnoprabg  6192  opabex3d  6354  opabex3  6355  eroveu  6904  dmaddpq  7752  dmmulpq  7753  prarloc  7876  ltexprlemopl  7974  ltexprlemlol  7975  ltexprlemopu  7976  ltexprlemupu  7977  shftdm  11609  fngzsum  13767  gzsumvalx  13768  ntreq0  15358  bdbm1.3ii  17122