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Theorem rabswap 2488
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 253 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴))
21abbii 2153 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ (𝑥𝐵𝑥𝐴)}
3 df-rab 2315 . 2 {𝑥𝐴𝑥𝐵} = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
4 df-rab 2315 . 2 {𝑥𝐵𝑥𝐴} = {𝑥 ∣ (𝑥𝐵𝑥𝐴)}
52, 3, 43eqtr4i 2070 1 {𝑥𝐴𝑥𝐵} = {𝑥𝐵𝑥𝐴}
Colors of variables: wff set class
Syntax hints:  wa 97   = wceq 1243  wcel 1393  {cab 2026  {crab 2310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-rab 2315
This theorem is referenced by: (None)
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