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Theorem abbi1dv 2740
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 2739 . 2 (𝜑𝐴 = {𝑥𝜓})
43eqcomd 2627 1 (𝜑 → {𝑥𝜓} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  {cab 2607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617
This theorem is referenced by:  abidnf  3357  csbtt  3525  csbie2g  3545  csbvarg  3975  iinxsng  4566  predep  5665  enfin2i  9087  fin1a2lem11  9176  hashf1  13179  shftuz  13743  psrbaglefi  19291  vmappw  24742  hdmap1fval  36566  hdmapfval  36599  hgmapfval  36658
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