Theorem r19.41v 2633

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 1528	. 2 |- F/ x ps
2	1	r19.41 2632	1 |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wrex 2456

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-rex 2461

This theorem is referenced by:  r19.42v  2634  3reeanv  2647  reuind  2942  iuncom4  3893  dfiun2g  3918  iunxiun  3968  inuni  4155  xpiundi  4684  xpiundir  4685  imaco  5134  coiun  5138  abrexco  5759  imaiun  5760  isoini  5818  rexrnmpo  5989  mapsnen  6810  genpassl  7522  genpassu  7523  4fvwrd4  10139  metrest  13976  trirec0xor  14763