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Theorem 2exeu 2398
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 2348 . . . 4  |-  ( E! x E. y ph  ->  E* x E. y ph )
2 euex 2343 . . . . 5  |-  ( E! y ph  ->  E. y ph )
32moimi 2369 . . . 4  |-  ( E* x E. y ph  ->  E* x E! y
ph )
41, 3syl 17 . . 3  |-  ( E! x E. y ph  ->  E* x E! y
ph )
5 2euex 2393 . . 3  |-  ( E! y E. x ph  ->  E. x E! y
ph )
64, 5anim12ci 577 . 2  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  -> 
( E. x E! y ph  /\  E* x E! y ph )
)
7 eu5 2345 . 2  |-  ( E! x E! y ph  <->  ( E. x E! y
ph  /\  E* x E! y ph ) )
86, 7sylibr 217 1  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E! y ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376   E.wex 1671   E!weu 2319   E*wmo 2320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-eu 2323  df-mo 2324
This theorem is referenced by:  2eu1  2402  2eu2  2403  2eu3  2404
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