Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2reu2 Unicode version

Theorem 2reu2 27874
Description: Double restricted existential uniqueness, analogous to 2eu2 2361. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
Assertion
Ref Expression
2reu2  |-  ( E! y  e.  B  E. x  e.  A  ph  ->  ( E! x  e.  A  E! y  e.  B  ph  <->  E! x  e.  A  E. y  e.  B  ph )
)
Distinct variable groups:    x, y, A    x, B
Allowed substitution hints:    ph( x, y)    B( y)

Proof of Theorem 2reu2
StepHypRef Expression
1 reurmo 2910 . . 3  |-  ( E! y  e.  B  E. x  e.  A  ph  ->  E* y  e.  B E. x  e.  A  ph )
2 2rmorex 3125 . . 3  |-  ( E* y  e.  B E. x  e.  A  ph  ->  A. x  e.  A  E* y  e.  B ph )
3 2reu1 27873 . . . 4  |-  ( A. 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  E. x  e.  A  ph ) ) )
4 simpl 444 . . . 4  |-  ( ( E! x  e.  A  E. y  e.  B  ph 
/\  E! y  e.  B  E. x  e.  A  ph )  ->  E! x  e.  A  E. y  e.  B  ph )
53, 4syl6bi 220 . . 3  |-  ( A. 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 )
)
61, 2, 53syl 19 . 2  |-  ( E! y  e.  B  E. x  e.  A  ph  ->  ( E! x  e.  A  E! y  e.  B  ph 
->  E! x  e.  A  E. y  e.  B  ph ) )
7 2rexreu 27872 . . 3  |-  ( ( E! x  e.  A  E. y  e.  B  ph 
/\  E! y  e.  B  E. x  e.  A  ph )  ->  E! x  e.  A  E! y  e.  B  ph )
87expcom 425 . 2  |-  ( E! y  e.  B  E. x  e.  A  ph  ->  ( E! x  e.  A  E. y  e.  B  ph 
->  E! x  e.  A  E! y  e.  B  ph ) )
96, 8impbid 184 1  |-  ( E! y  e.  B  E. x  e.  A  ph  ->  ( E! x  e.  A  E! y  e.  B  ph  <->  E! x  e.  A  E. y  e.  B  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wral 2692   E.wrex 2693   E!wreu 2694   E*wrmo 2695
This theorem is referenced by:  2reu8  27879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700
  Copyright terms: Public domain W3C validator