Theorem rabswap 2791

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 437	. . 3 |- ((x e. A /\ x e. B) <-> (x e. B /\ x e. A))
2	1	abbii 2466	. 2 |- {x | (x e. A /\ x e. B)} = {x | (x e. B /\ x e. A)}
3		df-rab 2624	. 2 |- {x e. A | x e. B} = {x | (x e. A /\ x e. B)}
4		df-rab 2624	. 2 |- {x e. B | x e. A} = {x | (x e. B /\ x e. A)}
5	2, 3, 4	3eqtr4i 2383	1 |- {x e. A | x e. B} = {x e. B | x e. A}

Syntax hints:   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339  {crab 2619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-rab 2624

This theorem is referenced by: (None)