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Theorem ralnex2 2616
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 2465 . . 3  |-  ( A. y  e.  B  -.  ph  <->  -. 
E. y  e.  B  ph )
21ralbii 2483 . 2  |-  ( A. x  e.  A  A. y  e.  B  -.  ph  <->  A. x  e.  A  -.  E. y  e.  B  ph )
3 ralnex 2465 . 2  |-  ( A. x  e.  A  -.  E. y  e.  B  ph  <->  -. 
E. x  e.  A  E. y  e.  B  ph )
42, 3bitri 184 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 105   A.wral 2455   E.wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie2 1494  ax-4 1510  ax-17 1526
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-ral 2460  df-rex 2461
This theorem is referenced by: (None)
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