Theorem r19.41v 2701

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 1577	. 2 |- F/ x ps
2	1	r19.41 2700	1 |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wrex 2523

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  df-nf 1510  df-rex 2528

This theorem is referenced by:  r19.42v  2702  3reeanv  2716  reuind  3024  iuncom4  4000  dfiun2g  4025  iunxiun  4075  inuni  4269  xpiundi  4810  xpiundir  4811  imaco  5270  coiun  5274  abrexco  5934  imaiun  5935  isoini  5993  rexrnmpo  6171  mapsnend  7054  mapsnen  7055  genpassl  7841  genpassu  7842  4fvwrd4  10478  4sqlem12  13104  metrest  15388  trirec0xor  16846