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Theorem abbi1dv 2259
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 1846 . 2 (𝜑 → ∀𝑥(𝜓𝑥𝐴))
3 abeq1 2249 . 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 1329   = wceq 1331  wcel 1480  {cab 2125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135
This theorem is referenced by:  abidnf  2852  csbtt  3014  csbvarg  3030  csbie2g  3050  abvor0dc  3386  iinxsng  3889  shftuz  10613
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