Theorem raaanv 3328

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 1421	. 2 |- F/ y ph
2		nfv 1421	. 2 |- F/ x ps
3	1, 2	raaan 3327	1 |- ( A. x e. A A. y e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. y e. A ps ) )

Syntax hints:   /\ wa 97   <-> wb 98   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311

This theorem is referenced by:  reusv3i  4191  f1mpt  5410