ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  excom13 Unicode version

Theorem excom13 1624
Description: Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
Assertion
Ref Expression
excom13  |-  ( E. x E. y E. z ph  <->  E. z E. y E. x ph )

Proof of Theorem excom13
StepHypRef Expression
1 excom 1599 . 2  |-  ( E. x E. y E. z ph  <->  E. y E. x E. z ph )
2 excom 1599 . . 3  |-  ( E. x E. z ph  <->  E. z E. x ph )
32exbii 1541 . 2  |-  ( E. y E. x E. z ph  <->  E. y E. z E. x ph )
4 excom 1599 . 2  |-  ( E. y E. z E. x ph  <->  E. z E. y E. x ph )
51, 3, 43bitri 204 1  |-  ( E. x E. y E. z ph  <->  E. z E. y E. x ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   E.wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  exrot3  1625  exrot4  1626  euotd  4081
  Copyright terms: Public domain W3C validator