Theorem raaanv 3567

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 1551	. 2 |- F/ y ph
2		nfv 1551	. 2 |- F/ x ps
3	1, 2	raaan 3566	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 2484

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489

This theorem is referenced by:  reusv3i  4507  f1mpt  5842