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Theorem rabbidv 2649
Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 10-Feb-1995.)
Hypothesis
Ref Expression
rabbidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
rabbidv (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem rabbidv
StepHypRef Expression
1 rabbidv.1 . . 3 (𝜑 → (𝜓𝜒))
21adantr 274 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32rabbidva 2648 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  wcel 1465  {crab 2397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-ral 2398  df-rab 2402
This theorem is referenced by:  rabeqbidv  2655  difeq2  3158  seex  4227  mptiniseg  5003  supeq1  6841  supeq2  6844  supeq3  6845  cardcl  7005  isnumi  7006  cardval3ex  7009  carden2bex  7013  genpdflem  7283  genipv  7285  genpelxp  7287  addcomprg  7354  mulcomprg  7356  uzval  9296  ixxval  9647  fzval  9760  hashinfom  10492  hashennn  10494  shftfn  10564  gcdval  11575  lcmval  11671  isprm  11717  istopon  12107  toponsspwpwg  12116  clsval  12207  neival  12239  cnpval  12294  blvalps  12484  blval  12485  limccl  12724  ellimc3apf  12725  eldvap  12747
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