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Theorem r2al 2431
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 2258 . 2  |-  F/_ y A
21r2alf 2429 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 1314    e. wcel 1465   A.wral 2393
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398
This theorem is referenced by:  r3al  2454  raliunxp  4650  codir  4897  qfto  4898  fununi  5161  dff13  5637  mpo2eqb  5848  qliftfun  6479  cnmpt12  12383  cnmpt22  12390
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