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Theorem 2reu5a 27955
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 2753 . 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 2753 . . . 4  |-  ( E! y  e.  B  ph  <->  ( E. y  e.  B  ph 
/\  E* y  e.  B ph ) )
32rexbii 2568 . . 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 2731 . . 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 678 . 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 240 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 176    /\ wa 358   E.wrex 2544   E!wreu 2545   E*wrmo 2546
This theorem is referenced by:  2reu1  27964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-rex 2549  df-reu 2550  df-rmo 2551
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