Theorem raaanv 3575

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 1552	. 2 |- F/ y ph
2		nfv 1552	. 2 |- F/ x ps
3	1, 2	raaan 3574	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 2486

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491

This theorem is referenced by:  reusv3i  4524  f1mpt  5863