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Theorem abbi1dv 2892
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
Hypothesis
Ref Expression
abbi1dv.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 abbi1dv.1 . . . 4 (𝜑 → (𝜓𝑥𝐴))
21bicomd 213 . . 3 (𝜑 → (𝑥𝐴𝜓))
32abbi2dv 2891 . 2 (𝜑𝐴 = {𝑥𝜓})
43eqcomd 2777 1 (𝜑 → {𝑥𝜓} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  {cab 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767
This theorem is referenced by:  abidnf  3527  csbtt  3693  csbie2g  3713  csbvarg  4148  iinxsng  4735  predep  5848  enfin2i  9349  fin1a2lem11  9438  hashf1  13443  shftuz  14017  psrbaglefi  19587  vmappw  25063  hdmap1fval  37604  hdmapfval  37635  hgmapfval  37694
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