Theorem rabswap 2731

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 266	. . 3 |- ( ( x e. A /\ x e. B ) <-> ( x e. B /\ x e. A ) )
2	1	abbii 2354	. 2 |- { x | ( x e. A /\ x e. B ) } = { x | ( x e. B /\ x e. A ) }
3		df-rab 2537	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
4		df-rab 2537	. 2 |- { x e. B | x e. A } = { x | ( x e. B /\ x e. A ) }
5	2, 3, 4	3eqtr4i 2269	1 |- { x e. A | x e. B } = { x e. B | x e. A }

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   {crab 2532

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-rab 2537

This theorem is referenced by: (None)