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Theorem 2eu2 2237
Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu2  |-  ( E! y E. x ph  ->  ( E! x E! y ph  <->  E! x E. y ph ) )

Proof of Theorem 2eu2
StepHypRef Expression
1 eumo 2196 . . 3  |-  ( E! y E. x ph  ->  E* y E. x ph )
2 2moex 2227 . . 3  |-  ( E* y E. x ph  ->  A. x E* y ph )
3 2eu1 2236 . . . 4  |-  ( A. x E* y ph  ->  ( E! x E! y
ph 
<->  ( E! x E. y ph  /\  E! y E. x ph )
) )
4 simpl 443 . . . 4  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E. y ph )
53, 4syl6bi 219 . . 3  |-  ( A. x E* y ph  ->  ( E! x E! y
ph  ->  E! x E. y ph ) )
61, 2, 53syl 18 . 2  |-  ( E! y E. x ph  ->  ( E! x E! y ph  ->  E! x E. y ph )
)
7 2exeu 2233 . . 3  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E! y ph )
87expcom 424 . 2  |-  ( E! y E. x ph  ->  ( E! x E. y ph  ->  E! x E! y ph ) )
96, 8impbid 183 1  |-  ( E! y E. x ph  ->  ( E! x E! y ph  <->  E! x E. y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531   E!weu 2156   E*wmo 2157
This theorem is referenced by:  2eu8  2243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161
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