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Theorem abidnf 3698
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 2971 . . . 4 (𝑥𝐴 → Ⅎ𝑥 𝑧𝐴)
32nf5rd 2189 . . 3 (𝑥𝐴 → (𝑧𝐴 → ∀𝑥 𝑧𝐴))
41, 3impbid2 227 . 2 (𝑥𝐴 → (∀𝑥 𝑧𝐴𝑧𝐴))
54abbi1dv 2957 1 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1528   = wceq 1530  wcel 2107  {cab 2804  wnfc 2966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968
This theorem is referenced by:  dedhb  3699  nfopd  4819  nfimad  5937  nffvd  6681  nfunidALT2  35991  nfunidALT  35992  nfopdALT  35993
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