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

Theorem 2reu2 28053
Description: Double restricted existential uniqueness, analogous to 2eu2 2369. (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 2932 . . 3  |-  ( E! y  e.  B  E. x  e.  A  ph  ->  E* y  e.  B E. x  e.  A  ph )
2 2rmorex 3147 . . 3  |-  ( E* y  e.  B E. x  e.  A  ph  ->  A. x  e.  A  E* y  e.  B ph )
3 2reu1 28052 . . . 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 445 . . . 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 221 . . 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 28051 . . 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 426 . 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 185 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 178    /\ wa 360   A.wral 2712   E.wrex 2713   E!wreu 2714   E*wrmo 2715
This theorem is referenced by:  2reu8  28058
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 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
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 1661  df-eu 2292  df-mo 2293  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720
  Copyright terms: Public domain W3C validator