ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uneq1 Unicode version

Theorem uneq1 3145
Description: Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
uneq1  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )

Proof of Theorem uneq1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2151 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
21orbi1d 740 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  \/  x  e.  C
)  <->  ( x  e.  B  \/  x  e.  C ) ) )
3 elun 3139 . . 3  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
4 elun 3139 . . 3  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
52, 3, 43bitr4g 221 . 2  |-  ( A  =  B  ->  (
x  e.  ( A  u.  C )  <->  x  e.  ( B  u.  C
) ) )
65eqrdv 2086 1  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 664    = wceq 1289    e. wcel 1438    u. cun 2995
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001
This theorem is referenced by:  uneq2  3146  uneq12  3147  uneq1i  3148  uneq1d  3151  prprc1  3545  uniprg  3663  unexb  4258  relresfld  4947  relcoi1  4949  rdgeq2  6119  xpiderm  6343  findcard2  6585  findcard2s  6586  unfiexmid  6608  bdunexb  11468  bj-unexg  11469
  Copyright terms: Public domain W3C validator