Theorem raaanv 3598

Description: Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)

Assertion

Ref	Expression
raaanv	|- ( A. x e. A A. y e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. y e. A ps ) )

Distinct variable groups:   ph, y   ps, x   x, y, A

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem raaanv

Step	Hyp	Ref	Expression
1		nfv 1574	. 2 |- F/ y ph
2		nfv 1574	. 2 |- F/ x ps
3	1, 2	raaan 3597	1 |- ( A. x e. A A. y e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. y e. A ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   A.wral 2508

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513

This theorem is referenced by:  reusv3i  4550  f1mpt  5895