Theorem rabswap 2612

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

Step	Hyp	Ref	Expression
1		ancom 264	. . 3 |- ( ( x e. A /\ x e. B ) <-> ( x e. B /\ x e. A ) )
2	1	abbii 2256	. 2 |- { x | ( x e. A /\ x e. B ) } = { x | ( x e. B /\ x e. A ) }
3		df-rab 2426	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
4		df-rab 2426	. 2 |- { x e. B | x e. A } = { x | ( x e. B /\ x e. A ) }
5	2, 3, 4	3eqtr4i 2171	1 |- { x e. A | x e. B } = { x e. B | x e. A }

Syntax hints:   /\ wa 103   = wceq 1332   e. wcel 1481   {cab 2126   {crab 2421

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-rab 2426

This theorem is referenced by: (None)