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Theorem abbi2dv 2296
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 1874 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝜓))
3 abeq2 2286 . 2 (𝐴 = {𝑥𝜓} ↔ ∀𝑥(𝑥𝐴𝜓))
42, 3sylibr 134 1 (𝜑𝐴 = {𝑥𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351   = wceq 1353  wcel 2148  {cab 2163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173
This theorem is referenced by:  sbab  2305  iftrue  3539  iffalse  3542  iniseg  5000  fncnvima2  5637  isoini  5818  dftpos3  6262  unfiexmid  6916  tgval3  13451  txrest  13669  cnblcld  13928
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