MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2exeu Structured version   Unicode version

Theorem 2exeu 2360
Description: Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)
Assertion
Ref Expression
2exeu  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E! y ph )

Proof of Theorem 2exeu
StepHypRef Expression
1 eumo 2323 . . . 4  |-  ( E! x E. y ph  ->  E* x E. y ph )
2 euex 2306 . . . . 5  |-  ( E! y ph  ->  E. y ph )
32moimi 2330 . . . 4  |-  ( E* x E. y ph  ->  E* x E! y
ph )
41, 3syl 16 . . 3  |-  ( E! x E. y ph  ->  E* x E! y
ph )
5 2euex 2355 . . 3  |-  ( E! y E. x ph  ->  E. x E! y
ph )
64, 5anim12ci 552 . 2  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  -> 
( E. x E! y ph  /\  E* x E! y ph )
)
7 eu5 2321 . 2  |-  ( E! x E! y ph  <->  ( E. x E! y
ph  /\  E* x E! y ph ) )
86, 7sylibr 205 1  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E! y ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551   E!weu 2283   E*wmo 2284
This theorem is referenced by:  2eu1  2363  2eu2  2364  2eu3  2365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
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 2287  df-mo 2288
  Copyright terms: Public domain W3C validator