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Theorem rabswap 2586
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 264 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  x  e.  A )
)
21abbii 2233 . 2  |-  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  { x  |  (
x  e.  B  /\  x  e.  A ) }
3 df-rab 2402 . 2  |-  { x  e.  A  |  x  e.  B }  =  {
x  |  ( x  e.  A  /\  x  e.  B ) }
4 df-rab 2402 . 2  |-  { x  e.  B  |  x  e.  A }  =  {
x  |  ( x  e.  B  /\  x  e.  A ) }
52, 3, 43eqtr4i 2148 1  |-  { x  e.  A  |  x  e.  B }  =  {
x  e.  B  |  x  e.  A }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316    e. wcel 1465   {cab 2103   {crab 2397
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-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-rab 2402
This theorem is referenced by: (None)
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