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Theorem 2reu5a 27969
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 2927 . 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 2927 . . . 4  |-  ( E! y  e.  B  ph  <->  ( E. y  e.  B  ph 
/\  E* y  e.  B ph ) )
32rexbii 2736 . . 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 2905 . . 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 680 . 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 242 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 178    /\ wa 360   E.wrex 2712   E!wreu 2713   E*wrmo 2714
This theorem is referenced by:  2reu1  27978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-rex 2717  df-reu 2718  df-rmo 2719
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