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Theorem rabbi2dva 3799
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 3563 . 2 (𝐴𝐵) = {𝑥𝐴𝑥𝐵}
2 rabbi2dva.1 . . 3 ((𝜑𝑥𝐴) → (𝑥𝐵𝜓))
32rabbidva 3176 . 2 (𝜑 → {𝑥𝐴𝑥𝐵} = {𝑥𝐴𝜓})
41, 3syl5eq 2667 1 (𝜑 → (𝐴𝐵) = {𝑥𝐴𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {crab 2911  cin 3554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-rab 2916  df-in 3562
This theorem is referenced by:  fndmdif  6277  bitsshft  15121  sylow3lem2  17964  leordtvallem1  20924  leordtvallem2  20925  ordtt1  21093  xkoccn  21332  txcnmpt  21337  xkopt  21368  ordthmeolem  21514  qustgphaus  21836  itg2monolem1  23423  lhop1  23681  efopn  24304  dirith  25118  pjvec  28404  pjocvec  28405  neibastop3  31999  diarnN  35898
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