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Theorem rabswap 3318
 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 453 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐵𝑥𝐴))
21abbii 2839 . 2 {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ (𝑥𝐵𝑥𝐴)}
3 df-rab 3092 . 2 {𝑥𝐴𝑥𝐵} = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
4 df-rab 3092 . 2 {𝑥𝐵𝑥𝐴} = {𝑥 ∣ (𝑥𝐵𝑥𝐴)}
52, 3, 43eqtr4i 2807 1 {𝑥𝐴𝑥𝐵} = {𝑥𝐵𝑥𝐴}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 387   = wceq 1508   ∈ wcel 2051  {cab 2753  {crab 3087 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-9 2060  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-sb 2017  df-clab 2754  df-cleq 2766  df-rab 3092 This theorem is referenced by: (None)
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