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Theorem rabswap 2669
Description: Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
Assertion
Ref Expression
rabswap  |-  { x  e.  A  |  x  e.  B }  =  {
x  e.  B  |  x  e.  A }

Proof of Theorem rabswap
StepHypRef Expression
1 ancom 266 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  x  e.  A )
)
21abbii 2305 . 2  |-  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  { x  |  (
x  e.  B  /\  x  e.  A ) }
3 df-rab 2477 . 2  |-  { x  e.  A  |  x  e.  B }  =  {
x  |  ( x  e.  A  /\  x  e.  B ) }
4 df-rab 2477 . 2  |-  { x  e.  B  |  x  e.  A }  =  {
x  |  ( x  e.  B  /\  x  e.  A ) }
52, 3, 43eqtr4i 2220 1  |-  { x  e.  A  |  x  e.  B }  =  {
x  e.  B  |  x  e.  A }
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1364    e. wcel 2160   {cab 2175   {crab 2472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-rab 2477
This theorem is referenced by: (None)
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