Theorem r19.41v 2510

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 1461	. 2 |- F/ x ps
2	1	r19.41 2509	1 |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) )

Syntax hints:   /\ wa 102   <-> wb 103   E.wrex 2349

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-rex 2354

This theorem is referenced by:  r19.42v  2511  3reeanv  2524  reuind  2795  iuncom4  3685  dfiun2g  3710  iunxiun  3757  inuni  3930  xpiundi  4416  xpiundir  4417  imaco  4846  coiun  4850  abrexco  5419  imaiun  5420  isoini  5477  rexrnmpt2  5636  genpassl  6714  genpassu  6715  4fvwrd4  9150