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Theorem r2al 2429
Description: Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
Assertion
Ref Expression
r2al  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. x A. y ( ( x  e.  A  /\  y  e.  B )  ->  ph )
)
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    A( x)    B( x, y)

Proof of Theorem r2al
StepHypRef Expression
1 nfcv 2256 . 2  |-  F/_ y A
21r2alf 2427 1  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. x A. y ( ( x  e.  A  /\  y  e.  B )  ->  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312    e. wcel 1463   A.wral 2391
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396
This theorem is referenced by:  r3al  2452  raliunxp  4648  codir  4895  qfto  4896  fununi  5159  dff13  5635  mpo2eqb  5846  qliftfun  6477  cnmpt12  12351  cnmpt22  12358
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