MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2eu2ex Unicode version

Theorem 2eu2ex 2217
Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu2ex  |-  ( E! x E! y ph  ->  E. x E. y ph )

Proof of Theorem 2eu2ex
StepHypRef Expression
1 euex 2166 . 2  |-  ( E! x E! y ph  ->  E. x E! y
ph )
2 euex 2166 . . 3  |-  ( E! y ph  ->  E. y ph )
32eximi 1563 . 2  |-  ( E. x E! y ph  ->  E. x E. y ph )
41, 3syl 15 1  |-  ( E! x E! y ph  ->  E. x E. y ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1528   E!weu 2143
This theorem is referenced by:  2eu1  2223
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
  Copyright terms: Public domain W3C validator