ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elequ1 Unicode version

Theorem elequ1 1641
Description: An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
elequ1  |-  ( x  =  y  ->  (
x  e.  z  <->  y  e.  z ) )

Proof of Theorem elequ1
StepHypRef Expression
1 ax-13 1445 . 2  |-  ( x  =  y  ->  (
x  e.  z  -> 
y  e.  z ) )
2 ax-13 1445 . . 3  |-  ( y  =  x  ->  (
y  e.  z  ->  x  e.  z )
)
32equcoms 1635 . 2  |-  ( x  =  y  ->  (
y  e.  z  ->  x  e.  z )
)
41, 3impbid 127 1  |-  ( x  =  y  ->  (
x  e.  z  <->  y  e.  z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103
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-gen 1379  ax-ie2 1424  ax-8 1436  ax-13 1445  ax-17 1460  ax-i9 1464
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  cleljust  1855  elsb3  1894  dveel1  1938  nalset  3916  zfpow  3957  mss  3989  zfun  4197  bj-nalset  10871  bj-nnelirr  10933
  Copyright terms: Public domain W3C validator