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Theorem abidnf 2797
 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 1453 . . 3 (∀𝑥 𝑧𝐴𝑧𝐴)
2 nfcr 2227 . . . 4 (𝑥𝐴 → Ⅎ𝑥 𝑧𝐴)
32nfrd 1465 . . 3 (𝑥𝐴 → (𝑧𝐴 → ∀𝑥 𝑧𝐴))
41, 3impbid2 142 . 2 (𝑥𝐴 → (∀𝑥 𝑧𝐴𝑧𝐴))
54abbi1dv 2214 1 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1294   = wceq 1296   ∈ wcel 1445  {cab 2081  Ⅎwnfc 2222 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224 This theorem is referenced by:  dedhb  2798  nfopd  3661  nfimad  4816  nffvd  5352
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