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Theorem abbi1dv 2232
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 1826 . 2 (𝜑 → ∀𝑥(𝜓𝑥𝐴))
3 abeq1 2222 . 2 ({𝑥𝜓} = 𝐴 ↔ ∀𝑥(𝜓𝑥𝐴))
42, 3sylibr 133 1 (𝜑 → {𝑥𝜓} = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1310   = wceq 1312  wcel 1461  {cab 2099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-11 1465  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109
This theorem is referenced by:  abidnf  2819  csbtt  2979  csbvarg  2994  csbie2g  3014  abvor0dc  3350  iinxsng  3850  shftuz  10476
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