Theorem rabswap 2712

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 2347	. 2 |- { x | ( x e. A /\ x e. B ) } = { x | ( x e. B /\ x e. A ) }
3		df-rab 2519	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
4		df-rab 2519	. 2 |- { x e. B | x e. A } = { x | ( x e. B /\ x e. A ) }
5	2, 3, 4	3eqtr4i 2262	1 |- { x e. A | x e. B } = { x e. B | x e. A }

Syntax hints:   /\ wa 104   = wceq 1397   e. wcel 2202   {cab 2217   {crab 2514

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-rab 2519

This theorem is referenced by: (None)