New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  rabid2 GIF version

Theorem rabid2 2788
 Description: An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
rabid2 (A = {x A φ} ↔ x A φ)
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem rabid2
StepHypRef Expression
1 abeq2 2458 . . 3 (A = {x (x A φ)} ↔ x(x A ↔ (x A φ)))
2 pm4.71 611 . . . 4 ((x Aφ) ↔ (x A ↔ (x A φ)))
32albii 1566 . . 3 (x(x Aφ) ↔ x(x A ↔ (x A φ)))
41, 3bitr4i 243 . 2 (A = {x (x A φ)} ↔ x(x Aφ))
5 df-rab 2623 . . 3 {x A φ} = {x (x A φ)}
65eqeq2i 2363 . 2 (A = {x A φ} ↔ A = {x (x A φ)})
7 df-ral 2619 . 2 (x A φx(x Aφ))
84, 6, 73bitr4i 268 1 (A = {x A φ} ↔ x A φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  {crab 2618 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-ral 2619  df-rab 2623 This theorem is referenced by:  rabxm  3573  iinrab2  4029  riinrab  4041  opeq  4619  dmmptg  5684  fmpt  5692
 Copyright terms: Public domain W3C validator