Theorem rabswap 2687

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 2323	. 2 |- { x | ( x e. A /\ x e. B ) } = { x | ( x e. B /\ x e. A ) }
3		df-rab 2495	. 2 |- { x e. A | x e. B } = { x | ( x e. A /\ x e. B ) }
4		df-rab 2495	. 2 |- { x e. B | x e. A } = { x | ( x e. B /\ x e. A ) }
5	2, 3, 4	3eqtr4i 2238	1 |- { x e. A | x e. B } = { x e. B | x e. A }

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2178   {cab 2193   {crab 2490

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-rab 2495

This theorem is referenced by: (None)