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Theorem rabbidv 2741
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 276 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32rabbidva 2740 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1364  wcel 2160  {crab 2472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-ral 2473  df-rab 2477
This theorem is referenced by:  rabeqbidv  2747  difeq2  3262  seex  4353  mptiniseg  5141  supeq1  7015  supeq2  7018  supeq3  7019  cardcl  7210  isnumi  7211  cardval3ex  7214  carden2bex  7218  genpdflem  7536  genipv  7538  genpelxp  7540  addcomprg  7607  mulcomprg  7609  uzval  9560  ixxval  9926  fzval  10040  hashinfom  10790  hashennn  10792  shftfn  10865  gcdval  11992  lcmval  12095  isprm  12141  odzval  12273  pceulem  12326  pceu  12327  pcval  12328  pczpre  12329  pcdiv  12334  lspval  13706  istopon  13973  toponsspwpwg  13982  clsval  14071  neival  14103  cnpval  14158  blvalps  14348  blval  14349  limccl  14588  ellimc3apf  14589  eldvap  14611
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