New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  abid2f GIF version

Theorem abid2f 2514
 Description: A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypothesis
Ref Expression
abid2f.1 xA
Assertion
Ref Expression
abid2f {x x A} = A

Proof of Theorem abid2f
StepHypRef Expression
1 abid2f.1 . . . . 5 xA
2 nfab1 2491 . . . . 5 x{x x A}
31, 2cleqf 2513 . . . 4 (A = {x x A} ↔ x(x Ax {x x A}))
4 abid 2341 . . . . . 6 (x {x x A} ↔ x A)
54bibi2i 304 . . . . 5 ((x Ax {x x A}) ↔ (x Ax A))
65albii 1566 . . . 4 (x(x Ax {x x A}) ↔ x(x Ax A))
73, 6bitri 240 . . 3 (A = {x x A} ↔ x(x Ax A))
8 biid 227 . . 3 (x Ax A)
97, 8mpgbir 1550 . 2 A = {x x A}
109eqcomi 2357 1 {x x A} = A
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476 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-or 359  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-nfc 2478 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator