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Theorem abidnf 3643
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 2174 . . 3 (∀𝑥 𝑧𝐴𝑧𝐴)
2 nfcr 2890 . . . 4 (𝑥𝐴 → Ⅎ𝑥 𝑧𝐴)
32nf5rd 2187 . . 3 (𝑥𝐴 → (𝑧𝐴 → ∀𝑥 𝑧𝐴))
41, 3impbid2 225 . 2 (𝑥𝐴 → (∀𝑥 𝑧𝐴𝑧𝐴))
54abbi1dv 2876 1 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1537   = wceq 1539  wcel 2104  {cab 2713  wnfc 2885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887
This theorem is referenced by:  dedhb  3644  nfopd  4826  nfimad  5984  nffvd  6812  nfunidALT2  37022  nfunidALT  37023  nfopdALT  37024
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