Theorem rabswap 1768

Description: Swap with a membership relation in a restricted class abstraction.

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 435	. . 3 |- ((x e. A /\ x e. B) <-> (x e. B /\ x e. A))
2	1	abbii 1572	. 2 |- {x | (x e. A /\ x e. B)} = {x | (x e. B /\ x e. A)}
3		df-rab 1649	. 2 |- {x e. A | x e. B} = {x | (x e. A /\ x e. B)}
4		df-rab 1649	. 2 |- {x e. B | x e. A} = {x | (x e. B /\ x e. A)}
5	2, 3, 4	3eqtr4 1502	1 |- {x e. A | x e. B} = {x e. B | x e. A}

Syntax hints:   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461  {crab 1645

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-rab 1649

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