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Theorem rabswap 2545
Description: Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
Assertion
Ref Expression
rabswap {𝑥𝐴𝑥𝐵} = {𝑥𝐵𝑥𝐴}

Proof of Theorem rabswap
StepHypRef Expression
1 ancom 262 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴))
21abbii 2203 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ (𝑥𝐵𝑥𝐴)}
3 df-rab 2368 . 2 {𝑥𝐴𝑥𝐵} = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
4 df-rab 2368 . 2 {𝑥𝐵𝑥𝐴} = {𝑥 ∣ (𝑥𝐵𝑥𝐴)}
52, 3, 43eqtr4i 2118 1 {𝑥𝐴𝑥𝐵} = {𝑥𝐵𝑥𝐴}
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wcel 1438  {cab 2074  {crab 2363
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-rab 2368
This theorem is referenced by: (None)
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