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Theorem rabbi2dva 4178
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 3913 . 2 (𝐴𝐵) = {𝑥𝐴𝑥𝐵}
2 rabbi2dva.1 . . 3 ((𝜑𝑥𝐴) → (𝑥𝐵𝜓))
32rabbidva 3422 . 2 (𝜑 → {𝑥𝐴𝑥𝐵} = {𝑥𝐴𝜓})
41, 3eqtrid 2810 1 (𝜑 → (𝐴𝐵) = {𝑥𝐴𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 400   = wceq 1570  wcel 2143  {crab 3416  cin 3904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735
This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-rab 3417  df-in 3912
This theorem is referenced by:  fndmdif  7037  bitsshft  16528  sylow3lem2  19693  leordtvallem1  23367  leordtvallem2  23368  ordtt1  23536  xkoccn  23776  txcnmpt  23781  xkopt  23812  ordthmeolem  23958  qustgphaus  24280  itg2monolem1  25909  lhop1  26173  efopn  26823  dirith  27693  pjvec  32048  pjocvec  32049  dfscott3  35512  neibastop3  36873  diarnN  41903
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