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

Theorem cleljust 1855
Description: When the class variables of set theory are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 1435 with the class variables in wcel 1434. (Contributed by NM, 28-Jan-2004.)
Assertion
Ref Expression
cleljust  |-  ( x  e.  y  <->  E. z
( z  =  x  /\  z  e.  y ) )
Distinct variable groups:    x, z    y,
z

Proof of Theorem cleljust
StepHypRef Expression
1 ax-17 1460 . . 3  |-  ( x  e.  y  ->  A. z  x  e.  y )
2 elequ1 1641 . . 3  |-  ( z  =  x  ->  (
z  e.  y  <->  x  e.  y ) )
31, 2equsex 1657 . 2  |-  ( E. z ( z  =  x  /\  z  e.  y )  <->  x  e.  y )
43bicomi 130 1  |-  ( x  e.  y  <->  E. z
( z  =  x  /\  z  e.  y ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   E.wex 1422
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-13 1445  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator