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Theorem rabswap 2732
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 437 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  x  e.  A )
)
21abbii 2408 . 2  |-  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  { x  |  (
x  e.  B  /\  x  e.  A ) }
3 df-rab 2565 . 2  |-  { x  e.  A  |  x  e.  B }  =  {
x  |  ( x  e.  A  /\  x  e.  B ) }
4 df-rab 2565 . 2  |-  { x  e.  B  |  x  e.  A }  =  {
x  |  ( x  e.  B  /\  x  e.  A ) }
52, 3, 43eqtr4i 2326 1  |-  { x  e.  A  |  x  e.  B }  =  {
x  e.  B  |  x  e.  A }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-rab 2565
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