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Theorem abidnf 3005
 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 z A} = A)
Distinct variable groups:   x,z   z,A
Allowed substitution hint:   A(x)

Proof of Theorem abidnf
StepHypRef Expression
1 sp 1747 . . 3 (x z Az A)
2 nfcr 2481 . . . 4 (xA → Ⅎx z A)
32nfrd 1763 . . 3 (xA → (z Ax z A))
41, 3impbid2 195 . 2 (xA → (x z Az A))
54abbi1dv 2469 1 (xA → {z x z A} = A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478 This theorem is referenced by:  dedhb  3006  nfopd  4605  nfimad  4954  nffvd  5335
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