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Theorem r2alf 2779
Description: Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 2778. (Revised by Wolf Lammen, 9-Jan-2020.)
Hypothesis
Ref Expression
r2alf.1  |-  F/_ y A
Assertion
Ref Expression
r2alf  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. x A. y ( ( x  e.  A  /\  y  e.  B )  ->  ph )
)
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)    B( x, y)

Proof of Theorem r2alf
StepHypRef Expression
1 r2alf.1 . . . 4  |-  F/_ y A
21nfcri 2557 . . 3  |-  F/ y  x  e.  A
3219.21 1933 . 2  |-  ( A. y ( x  e.  A  ->  ( y  e.  B  ->  ph )
)  <->  ( x  e.  A  ->  A. y
( y  e.  B  ->  ph ) ) )
43r2allem 2778 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1403    e. wcel 1842   F/_wnfc 2550   A.wral 2753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1634  df-nf 1638  df-sb 1764  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758
This theorem is referenced by:  r2alOLD  2782  r2exf  2927  ralcomf  2965
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