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Theorem 2reu5a 27826
Description: Double restricted existential uniqueness in terms of restricted existence and restricted "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2reu5a  |-  ( E! x  e.  A  E! y  e.  B  ph  <->  ( E. x  e.  A  ( E. y  e.  B  ph 
/\  E* y  e.  B ph )  /\  E* x  e.  A ( E. y  e.  B  ph  /\  E* y  e.  B ph ) ) )

Proof of Theorem 2reu5a
StepHypRef Expression
1 reu5 2885 . 2  |-  ( E! x  e.  A  E! y  e.  B  ph  <->  ( E. x  e.  A  E! y  e.  B  ph  /\  E* x  e.  A E! y  e.  B  ph ) )
2 reu5 2885 . . . 4  |-  ( E! y  e.  B  ph  <->  ( E. y  e.  B  ph 
/\  E* y  e.  B ph ) )
32rexbii 2695 . . 3  |-  ( E. x  e.  A  E! y  e.  B  ph  <->  E. x  e.  A  ( E. y  e.  B  ph  /\  E* y  e.  B ph ) )
42rmobii 2863 . . 3  |-  ( E* x  e.  A E! y  e.  B  ph  <->  E* x  e.  A ( E. y  e.  B  ph 
/\  E* y  e.  B ph ) )
53, 4anbi12i 679 . 2  |-  ( ( E. x  e.  A  E! y  e.  B  ph 
/\  E* x  e.  A E! y  e.  B  ph )  <->  ( E. x  e.  A  ( E. y  e.  B  ph  /\  E* y  e.  B ph )  /\  E* x  e.  A ( E. y  e.  B  ph  /\  E* y  e.  B ph ) ) )
61, 5bitri 241 1  |-  ( E! x  e.  A  E! y  e.  B  ph  <->  ( E. x  e.  A  ( E. y  e.  B  ph 
/\  E* y  e.  B ph )  /\  E* x  e.  A ( E. y  e.  B  ph  /\  E* y  e.  B ph ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wrex 2671   E!wreu 2672   E*wrmo 2673
This theorem is referenced by:  2reu1  27835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-rex 2676  df-reu 2677  df-rmo 2678
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