Theorem raaanv 3470

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 1508	. 2 |- F/ y ph
2		nfv 1508	. 2 |- F/ x ps
3	1, 2	raaan 3469	1 |- ( A. x e. A A. y e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. y e. A ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wral 2416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421

This theorem is referenced by:  reusv3i  4380  f1mpt  5672