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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  16420  sylow3lem2  19537  leordtvallem1  22934  leordtvallem2  22935  ordtt1  23103  xkoccn  23343  txcnmpt  23348  xkopt  23379  ordthmeolem  23525  qustgphaus  23847  itg2monolem1  25492  lhop1  25755  efopn  26390  dirith  27256  pjvec  31204  pjocvec  31205  neibastop3  35550  diarnN  40303
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