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Theorem abbi2dv 2156
 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 1754 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝜓))
3 abeq2 2146 . 2 (𝐴 = {𝑥𝜓} ↔ ∀𝑥(𝑥𝐴𝜓))
42, 3sylibr 137 1 (𝜑𝐴 = {𝑥𝜓})
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ wcel 1393  {cab 2026 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036 This theorem is referenced by:  sbab  2164  iftrue  3336  iffalse  3339  iniseg  4697  fncnvima2  5288  isoini  5457  dftpos3  5877
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