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Theorem abbi2dv 2172
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 1770 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝜓))
3 abeq2 2162 . 2 (𝐴 = {𝑥𝜓} ↔ ∀𝑥(𝑥𝐴𝜓))
42, 3sylibr 141 1 (𝜑𝐴 = {𝑥𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = wceq 1259  wcel 1409  {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052
This theorem is referenced by:  sbab  2180  iftrue  3364  iffalse  3367  iniseg  4725  fncnvima2  5316  isoini  5485  dftpos3  5908
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