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Theorem rabswap 2546
 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 263 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴))
21abbii 2204 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ (𝑥𝐵𝑥𝐴)}
3 df-rab 2369 . 2 {𝑥𝐴𝑥𝐵} = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
4 df-rab 2369 . 2 {𝑥𝐵𝑥𝐴} = {𝑥 ∣ (𝑥𝐵𝑥𝐴)}
52, 3, 43eqtr4i 2119 1 {𝑥𝐴𝑥𝐵} = {𝑥𝐵𝑥𝐴}
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1290   ∈ wcel 1439  {cab 2075  {crab 2364 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-rab 2369 This theorem is referenced by: (None)
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