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Theorem abidnf 3362
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 2051 . . 3 (∀𝑥 𝑧𝐴𝑧𝐴)
2 nfcr 2753 . . . 4 (𝑥𝐴 → Ⅎ𝑥 𝑧𝐴)
32nf5rd 2064 . . 3 (𝑥𝐴 → (𝑧𝐴 → ∀𝑥 𝑧𝐴))
41, 3impbid2 216 . 2 (𝑥𝐴 → (∀𝑥 𝑧𝐴𝑧𝐴))
54abbi1dv 2740 1 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wnfc 2748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750
This theorem is referenced by:  dedhb  3363  nfopd  4394  nfimad  5444  nffvd  6167  nfunidALT2  33775  nfunidALT  33776  nfopdALT  33777
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