ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  difeq2 Unicode version

Theorem difeq2 3188
Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difeq2  |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B
) )

Proof of Theorem difeq2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2203 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
21notbid 656 . . 3  |-  ( A  =  B  ->  ( -.  x  e.  A  <->  -.  x  e.  B ) )
32rabbidv 2675 . 2  |-  ( A  =  B  ->  { x  e.  C  |  -.  x  e.  A }  =  { x  e.  C  |  -.  x  e.  B } )
4 dfdif2 3079 . 2  |-  ( C 
\  A )  =  { x  e.  C  |  -.  x  e.  A }
5 dfdif2 3079 . 2  |-  ( C 
\  B )  =  { x  e.  C  |  -.  x  e.  B }
63, 4, 53eqtr4g 2197 1  |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1331    e. wcel 1480   {crab 2420    \ cdif 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-rab 2425  df-dif 3073
This theorem is referenced by:  difeq12  3189  difeq2i  3191  difeq2d  3194  disjdif2  3441  ssdifeq0  3445  2oconcl  6336  diffitest  6781  diffifi  6788  undifdc  6812  sbthlem2  6846  isbth  6855  difinfinf  6986  ismkvnex  7029  iscld  12282
  Copyright terms: Public domain W3C validator