Theorem r19.41v 2707

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 1581	. 2 |- F/ x ps
2	1	r19.41 2706	1 |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-nf 1514  df-rex 2534

This theorem is used by:  r19.42v  2708  3reeanv  2722  reuind  3031  iuncom4  4019  dfiun2g  4044  iunxiun  4094  inuni  4291  xpiundi  4833  xpiundir  4834  imaco  5293  coiun  5297  abrexco  5965  imaiun  5966  isoini  6024  rexrnmpo  6204  mapsnend  7099  mapsnen  7100  genpassl  7891  genpassu  7892  4fvwrd4  10547  4sqlem12  13181  metrest  15607  trirec0xor  17094