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Theorem abbi2dv 2259
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 1847 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝜓))
3 abeq2 2249 . 2 (𝐴 = {𝑥𝜓} ↔ ∀𝑥(𝑥𝐴𝜓))
42, 3sylibr 133 1 (𝜑𝐴 = {𝑥𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1330   = wceq 1332  wcel 1481  {cab 2126
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136
This theorem is referenced by:  sbab  2268  iftrue  3483  iffalse  3486  iniseg  4918  fncnvima2  5548  isoini  5726  dftpos3  6166  unfiexmid  6813  tgval3  12264  txrest  12482  cnblcld  12741
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