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

Theorem 2euswap 2338
Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
Assertion
Ref Expression
2euswap  |-  ( A. x E* y ph  ->  ( E! x E. y ph  ->  E! y E. x ph ) )

Proof of Theorem 2euswap
StepHypRef Expression
1 excomim 1753 . . . 4  |-  ( E. x E. y ph  ->  E. y E. x ph )
21a1i 11 . . 3  |-  ( A. x E* y ph  ->  ( E. x E. y ph  ->  E. y E. x ph ) )
3 2moswap 2337 . . 3  |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) )
42, 3anim12d 547 . 2  |-  ( A. x E* y ph  ->  ( ( E. x E. y ph  /\  E* x E. y ph )  -> 
( E. y E. x ph  /\  E* y E. x ph )
) )
5 eu5 2300 . 2  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
6 eu5 2300 . 2  |-  ( E! y E. x ph  <->  ( E. y E. x ph  /\  E* y E. x ph ) )
74, 5, 63imtr4g 262 1  |-  ( A. x E* y ph  ->  ( E! x E. y ph  ->  E! y E. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547   E!weu 2262   E*wmo 2263
This theorem is referenced by:  euxfr2  3087  2reuswap  3104  2reuswap2  23936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267
  Copyright terms: Public domain W3C validator