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Theorem abbi2dv 2236
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
Hypothesis
Ref Expression
abbirdv.1 (𝜑 → (𝑥𝐴𝜓))
Assertion
Ref Expression
abbi2dv (𝜑𝐴 = {𝑥𝜓})
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem abbi2dv
StepHypRef Expression
1 abbirdv.1 . . 3 (𝜑 → (𝑥𝐴𝜓))
21alrimiv 1830 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝜓))
3 abeq2 2226 . 2 (𝐴 = {𝑥𝜓} ↔ ∀𝑥(𝑥𝐴𝜓))
42, 3sylibr 133 1 (𝜑𝐴 = {𝑥𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1314   = wceq 1316  wcel 1465  {cab 2103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113
This theorem is referenced by:  sbab  2244  iftrue  3449  iffalse  3452  iniseg  4881  fncnvima2  5509  isoini  5687  dftpos3  6127  unfiexmid  6774  tgval3  12154  txrest  12372  cnblcld  12631
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