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Theorem rabbi2dva 4217
Description: Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
Hypothesis
Ref Expression
rabbi2dva.1 ((𝜑𝑥𝐴) → (𝑥𝐵𝜓))
Assertion
Ref Expression
rabbi2dva (𝜑 → (𝐴𝐵) = {𝑥𝐴𝜓})
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rabbi2dva
StepHypRef Expression
1 dfin5 3956 . 2 (𝐴𝐵) = {𝑥𝐴𝑥𝐵}
2 rabbi2dva.1 . . 3 ((𝜑𝑥𝐴) → (𝑥𝐵𝜓))
32rabbidva 3439 . 2 (𝜑 → {𝑥𝐴𝑥𝐵} = {𝑥𝐴𝜓})
41, 3eqtrid 2784 1 (𝜑 → (𝐴𝐵) = {𝑥𝐴𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {crab 3432  cin 3947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-rab 3433  df-in 3955
This theorem is referenced by:  fndmdif  7043  bitsshft  16418  sylow3lem2  19498  leordtvallem1  22721  leordtvallem2  22722  ordtt1  22890  xkoccn  23130  txcnmpt  23135  xkopt  23166  ordthmeolem  23312  qustgphaus  23634  itg2monolem1  25275  lhop1  25538  efopn  26173  dirith  27039  pjvec  30987  pjocvec  30988  neibastop3  35333  diarnN  40086
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