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Theorem rabbi2dva 4191
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 3941 . 2 (𝐴𝐵) = {𝑥𝐴𝑥𝐵}
2 rabbi2dva.1 . . 3 ((𝜑𝑥𝐴) → (𝑥𝐵𝜓))
32rabbidva 3476 . 2 (𝜑 → {𝑥𝐴𝑥𝐵} = {𝑥𝐴𝜓})
41, 3syl5eq 2865 1 (𝜑 → (𝐴𝐵) = {𝑥𝐴𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {crab 3139  cin 3932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-rab 3144  df-in 3940
This theorem is referenced by:  fndmdif  6804  bitsshft  15812  sylow3lem2  18682  leordtvallem1  21746  leordtvallem2  21747  ordtt1  21915  xkoccn  22155  txcnmpt  22160  xkopt  22191  ordthmeolem  22337  qustgphaus  22658  itg2monolem1  24278  lhop1  24538  efopn  25168  dirith  26032  pjvec  29400  pjocvec  29401  neibastop3  33607  diarnN  38145
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