Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  abidnf Structured version   Visualization version   GIF version

Theorem abidnf 3342
 Description: Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.)
Assertion
Ref Expression
abidnf (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
Distinct variable groups:   𝑥,𝑧   𝑧,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem abidnf
StepHypRef Expression
1 sp 2041 . . 3 (∀𝑥 𝑧𝐴𝑧𝐴)
2 nfcr 2743 . . . 4 (𝑥𝐴 → Ⅎ𝑥 𝑧𝐴)
32nf5rd 2054 . . 3 (𝑥𝐴 → (𝑧𝐴 → ∀𝑥 𝑧𝐴))
41, 3impbid2 215 . 2 (𝑥𝐴 → (∀𝑥 𝑧𝐴𝑧𝐴))
54abbi1dv 2730 1 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  Ⅎwnfc 2738 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-nfc 2740 This theorem is referenced by:  dedhb  3343  nfopd  4357  nfimad  5394  nffvd  6112  nfunidALT2  33274  nfunidALT  33275  nfopdALT  33276
 Copyright terms: Public domain W3C validator