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Theorem 2exeu 2220
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 2183 . . . 4  |-  ( E! x E. y ph  ->  E* x E. y ph )
2 euex 2166 . . . . 5  |-  ( E! y ph  ->  E. y ph )
32moimi 2190 . . . 4  |-  ( E* x E. y ph  ->  E* x E! y
ph )
41, 3syl 15 . . 3  |-  ( E! x E. y ph  ->  E* x E! y
ph )
5 2euex 2215 . . 3  |-  ( E! y E. x ph  ->  E. x E! y
ph )
64, 5anim12ci 550 . 2  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  -> 
( E. x E! y ph  /\  E* x E! y ph )
)
7 eu5 2181 . 2  |-  ( E! x E! y ph  <->  ( E. x E! y
ph  /\  E* x E! y ph ) )
86, 7sylibr 203 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 358   E.wex 1528   E!weu 2143   E*wmo 2144
This theorem is referenced by:  2eu1  2223  2eu2  2224  2eu3  2225
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
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