Theorem 19.42v 1955

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 1575	. 2 |- ( ph -> A. x ph )
2	1	19.42h 1735	1 |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  exdistr  1958  19.42vv  1960  19.42vvv  1961  4exdistr  1965  cbvex2  1971  2sb5  2036  2sb5rf  2042  rexcom4a  2828  ceqsex2  2845  reuind  3012  2rmorex  3013  sbccomlem  3107  bm1.3ii  4215  opm  4332  eqvinop  4341  uniuni  4554  elco  4902  dmopabss  4949  dmopab3  4950  mptpreima  5237  brprcneu  5641  relelfvdm  5680  fndmin  5763  fliftf  5950  dfoprab2  6078  dmoprab  6112  dmoprabss  6113  fnoprabg  6132  opabex3d  6292  opabex3  6293  eroveu  6838  dmaddpq  7659  dmmulpq  7660  prarloc  7783  ltexprlemopl  7881  ltexprlemlol  7882  ltexprlemopu  7883  ltexprlemupu  7884  shftdm  11462  fngsum  13551  igsumvalx  13552  ntreq0  14943  bdbm1.3ii  16607