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Theorem rabswap 3098
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 465 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴))
21abbii 2726 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ (𝑥𝐵𝑥𝐴)}
3 df-rab 2905 . 2 {𝑥𝐴𝑥𝐵} = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
4 df-rab 2905 . 2 {𝑥𝐵𝑥𝐴} = {𝑥 ∣ (𝑥𝐵𝑥𝐴)}
52, 3, 43eqtr4i 2642 1 {𝑥𝐴𝑥𝐵} = {𝑥𝐵𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  wcel 1977  {cab 2596  {crab 2900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-rab 2905
This theorem is referenced by: (None)
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