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Theorem 2euex 2315
Description: Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
2euex  |-  ( E! x E. y ph  ->  E. y E! x ph )

Proof of Theorem 2euex
StepHypRef Expression
1 eu5 2264 . 2  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
2 excom 1871 . . . 4  |-  ( E. x E. y ph  <->  E. y E. x ph )
3 nfe1 1862 . . . . . 6  |-  F/ y E. y ph
43nfmo 2255 . . . . 5  |-  F/ y E* x E. y ph
5 19.8a 1879 . . . . . . 7  |-  ( ph  ->  E. y ph )
65moimi 2290 . . . . . 6  |-  ( E* x E. y ph  ->  E* x ph )
7 df-mo 2241 . . . . . 6  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
86, 7sylib 196 . . . . 5  |-  ( E* x E. y ph  ->  ( E. x ph  ->  E! x ph )
)
94, 8eximd 1904 . . . 4  |-  ( E* x E. y ph  ->  ( E. y E. x ph  ->  E. y E! x ph ) )
102, 9syl5bi 217 . . 3  |-  ( E* x E. y ph  ->  ( E. x E. y ph  ->  E. y E! x ph ) )
1110impcom 428 . 2  |-  ( ( E. x E. y ph  /\  E* x E. y ph )  ->  E. y E! x ph )
121, 11sylbi 195 1  |-  ( E! x E. y ph  ->  E. y E! x ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   E.wex 1631   E!weu 2236   E*wmo 2237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1406  df-ex 1632  df-nf 1636  df-eu 2240  df-mo 2241
This theorem is referenced by:  2exeu  2320
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