Theorem rabswap 2532

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 262	. . 3 |- ( ( x e. A /\ x e. B ) <-> ( x e. B /\ x e. A ) )
2	1	abbii 2194	. 2 |- { x | ( x e. A /\ x e. B ) } = { x | ( x e. B /\ x e. A ) }
3		df-rab 2357	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
4		df-rab 2357	. 2 |- { x e. B | x e. A } = { x | ( x e. B /\ x e. A ) }
5	2, 3, 4	3eqtr4i 2111	1 |- { x e. A | x e. B } = { x e. B | x e. A }

Syntax hints:   /\ wa 102   = wceq 1284   e. wcel 1433   {cab 2067   {crab 2352

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-rab 2357

This theorem is referenced by: (None)