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Theorem eqabcdv 2874
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
eqabcdv.1 (𝜑 → (𝜓𝑥𝐴))
Assertion
Ref Expression
eqabcdv (𝜑 → {𝑥𝜓} = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem eqabcdv
StepHypRef Expression
1 eqabcdv.1 . . . 4 (𝜑 → (𝜓𝑥𝐴))
21bicomd 223 . . 3 (𝜑 → (𝑥𝐴𝜓))
32eqabdv 2873 . 2 (𝜑𝐴 = {𝑥𝜓})
43eqcomd 2741 1 (𝜑 → {𝑥𝜓} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  {cab 2712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814
This theorem is referenced by:  abidnf  3711  csbtt  3925  csbie2g  3951  csbvarg  4440  iinxsng  5093  predep  6353  fnsnfv  6988  enfin2i  10359  fin1a2lem11  10448  hashf1  14493  shftuz  15105  psrbaglefi  21964  vmappw  27174  addsrid  28012  mulsrid  28154  hdmap1fval  41779  hdmapfval  41810  hgmapfval  41869
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