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

Theorem excomim 1787
Description: One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
Assertion
Ref Expression
excomim  |-  ( E. x E. y ph  ->  E. y E. x ph )

Proof of Theorem excomim
StepHypRef Expression
1 19.8a 1720 . . 3  |-  ( ph  ->  E. x ph )
212eximi 1566 . 2  |-  ( E. x E. y ph  ->  E. x E. y E. x ph )
3 nfe1 1708 . . . 4  |-  F/ x E. x ph
43nfex 1769 . . 3  |-  F/ x E. y E. x ph
5419.9 1785 . 2  |-  ( E. x E. y E. x ph  <->  E. y E. x ph )
62, 5sylib 188 1  |-  ( E. x E. y ph  ->  E. y E. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1530
This theorem is referenced by:  excom  1788  2euswap  2221  a9e2eq  28379  a9e2nd  28380  a9e2eqVD  28756  a9e2ndVD  28757  a9e2ndALT  28780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717
This theorem depends on definitions:  df-bi 177  df-tru 1310  df-ex 1531  df-nf 1534
  Copyright terms: Public domain W3C validator