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Theorem rabbi2dva 4179
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 3914 . 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 3905
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 3913
This theorem is referenced by:  fndmdif  7039  bitsshft  16534  sylow3lem2  19699  leordtvallem1  23348  leordtvallem2  23349  ordtt1  23517  xkoccn  23757  txcnmpt  23762  xkopt  23793  ordthmeolem  23939  qustgphaus  24261  itg2monolem1  25890  lhop1  26154  efopn  26804  dirith  27674  pjvec  32029  pjocvec  32030  dfscott3  35493  neibastop3  36854  diarnN  41884
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