Theorem r19.41v 2523

Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)

Assertion

Ref	Expression
r19.41v	|- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) )

Distinct variable group:   ps, x

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem r19.41v

Step	Hyp	Ref	Expression
1		nfv 1466	. 2 |- F/ x ps
2	1	r19.41 2522	1 |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) )

Syntax hints:   /\ wa 102   <-> wb 103   E.wrex 2360

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-rex 2365

This theorem is referenced by:  r19.42v  2524  3reeanv  2537  reuind  2820  iuncom4  3735  dfiun2g  3760  iunxiun  3808  inuni  3989  xpiundi  4492  xpiundir  4493  imaco  4931  coiun  4935  abrexco  5530  imaiun  5531  isoini  5589  rexrnmpt2  5752  mapsnen  6518  genpassl  7073  genpassu  7074  4fvwrd4  9539