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Theorem ralnex2 2572
Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 18-May-2023.)
Assertion
Ref Expression
ralnex2  |-  ( A. x  e.  A  A. y  e.  B  -.  ph  <->  -. 
E. x  e.  A  E. y  e.  B  ph )

Proof of Theorem ralnex2
StepHypRef Expression
1 ralnex 2427 . . 3  |-  ( A. y  e.  B  -.  ph  <->  -. 
E. y  e.  B  ph )
21ralbii 2442 . 2  |-  ( A. x  e.  A  A. y  e.  B  -.  ph  <->  A. x  e.  A  -.  E. y  e.  B  ph )
3 ralnex 2427 . 2  |-  ( A. x  e.  A  -.  E. y  e.  B  ph  <->  -. 
E. x  e.  A  E. y  e.  B  ph )
42, 3bitri 183 1  |-  ( A. x  e.  A  A. y  e.  B  -.  ph  <->  -. 
E. x  e.  A  E. y  e.  B  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 104   A.wral 2417   E.wrex 2418
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-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie2 1471  ax-4 1488  ax-17 1507
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-ral 2422  df-rex 2423
This theorem is referenced by: (None)
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