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Theorem abbi1dv 2173
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 1770 . 2 (𝜑 → ∀𝑥(𝜓𝑥𝐴))
3 abeq1 2163 . 2 ({𝑥𝜓} = 𝐴 ↔ ∀𝑥(𝜓𝑥𝐴))
42, 3sylibr 141 1 (𝜑 → {𝑥𝜓} = 𝐴)
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:  abidnf  2732  csbtt  2890  csbvarg  2905  csbie2g  2924  abvor0dc  3270  iinxsng  3758  shftuz  9646
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