Theorem r19.41v 2664

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 1552	. 2 |- F/ x ps
2	1	r19.41 2663	1 |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wrex 2487

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-rex 2492

This theorem is referenced by:  r19.42v  2665  3reeanv  2679  reuind  2985  iuncom4  3948  dfiun2g  3973  iunxiun  4023  inuni  4215  xpiundi  4751  xpiundir  4752  imaco  5207  coiun  5211  abrexco  5851  imaiun  5852  isoini  5910  rexrnmpo  6084  mapsnen  6927  genpassl  7672  genpassu  7673  4fvwrd4  10297  4sqlem12  12840  metrest  15093  trirec0xor  16186