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Theorem rabsn 3790
 Description: Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
Assertion
Ref Expression
rabsn (B A → {x A x = B} = {B})
Distinct variable groups:   x,A   x,B

Proof of Theorem rabsn
StepHypRef Expression
1 eleq1 2413 . . . . 5 (x = B → (x AB A))
21pm5.32ri 619 . . . 4 ((x A x = B) ↔ (B A x = B))
32baib 871 . . 3 (B A → ((x A x = B) ↔ x = B))
43abbidv 2467 . 2 (B A → {x (x A x = B)} = {x x = B})
5 df-rab 2623 . 2 {x A x = B} = {x (x A x = B)}
6 df-sn 3741 . 2 {B} = {x x = B}
74, 5, 63eqtr4g 2410 1 (B A → {x A x = B} = {B})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-rab 2623  df-sn 3741 This theorem is referenced by: (None)
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