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Theorem abbi1dv 2207
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
Hypothesis
Ref Expression
abbildv.1 (𝜑 → (𝜓𝑥𝐴))
Assertion
Ref Expression
abbi1dv (𝜑 → {𝑥𝜓} = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem abbi1dv
StepHypRef Expression
1 abbildv.1 . . 3 (𝜑 → (𝜓𝑥𝐴))
21alrimiv 1802 . 2 (𝜑 → ∀𝑥(𝜓𝑥𝐴))
3 abeq1 2197 . 2 ({𝑥𝜓} = 𝐴 ↔ ∀𝑥(𝜓𝑥𝐴))
42, 3sylibr 132 1 (𝜑 → {𝑥𝜓} = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287   = wceq 1289  wcel 1438  {cab 2074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084
This theorem is referenced by:  abidnf  2781  csbtt  2941  csbvarg  2956  csbie2g  2976  abvor0dc  3304  iinxsng  3798  shftuz  10216
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