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Theorem 2reu5a 37998
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 3051 . 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 3051 . . . 4  |-  ( E! y  e.  B  ph  <->  ( E. y  e.  B  ph 
/\  E* y  e.  B  ph ) )
32rexbii 2934 . . 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 3027 . . 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 701 . 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 252 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 setvar class
Syntax hints:    <-> wb 187    /\ wa 370   E.wrex 2783   E!wreu 2784   E*wrmo 2785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-12 1907
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-eu 2270  df-mo 2271  df-rex 2788  df-reu 2789  df-rmo 2790
This theorem is referenced by:  2reu1  38007
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