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Theorem ralnex2 2929
Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
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 notnot 292 . 2  |-  ( A. x  e.  A  A. y  e.  B  -.  ph  <->  -. 
-.  A. x  e.  A  A. y  e.  B  -.  ph )
2 notnot 292 . . . 4  |-  ( ph  <->  -. 
-.  ph )
322rexbii 2926 . . 3  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x  e.  A  E. y  e.  B  -.  -.  ph )
4 rexnal2 2927 . . 3  |-  ( E. x  e.  A  E. y  e.  B  -.  -.  ph  <->  -.  A. x  e.  A  A. y  e.  B  -.  ph )
53, 4bitr2i 253 . 2  |-  ( -. 
A. x  e.  A  A. y  e.  B  -.  ph  <->  E. x  e.  A  E. y  e.  B  ph )
61, 5xchbinx 311 1  |-  ( A. x  e.  A  A. y  e.  B  -.  ph  <->  -. 
E. x  e.  A  E. y  e.  B  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187   A.wral 2773   E.wrex 2774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-ral 2778  df-rex 2779
This theorem is referenced by:  axtgupdim2  24496  fourierdlem42  37799  fourierdlem42OLD  37800
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