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

Theorem 2euex 2215
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 2181 . 2  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
2 excom 1786 . . . 4  |-  ( E. x E. y ph  <->  E. y E. x ph )
3 nfe1 1706 . . . . . 6  |-  F/ y E. y ph
43nfmo 2160 . . . . 5  |-  F/ y E* x E. y ph
5 19.8a 1718 . . . . . . 7  |-  ( ph  ->  E. y ph )
65moimi 2190 . . . . . 6  |-  ( E* x E. y ph  ->  E* x ph )
7 df-mo 2148 . . . . . 6  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
86, 7sylib 188 . . . . 5  |-  ( E* x E. y ph  ->  ( E. x ph  ->  E! x ph )
)
94, 8eximd 1750 . . . 4  |-  ( E* x E. y ph  ->  ( E. y E. x ph  ->  E. y E! x ph ) )
102, 9syl5bi 208 . . 3  |-  ( E* x E. y ph  ->  ( E. x E. y ph  ->  E. y E! x ph ) )
1110impcom 419 . 2  |-  ( ( E. x E. y ph  /\  E* x E. y ph )  ->  E. y E! x ph )
121, 11sylbi 187 1  |-  ( E! x E. y ph  ->  E. y E! x 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:  2exeu  2220
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
  Copyright terms: Public domain W3C validator