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Theorem rabbidv 2727
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 2726 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  {crab 2459
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-ral 2460  df-rab 2464
This theorem is referenced by:  rabeqbidv  2733  difeq2  3248  seex  4336  mptiniseg  5124  supeq1  6985  supeq2  6988  supeq3  6989  cardcl  7180  isnumi  7181  cardval3ex  7184  carden2bex  7188  genpdflem  7506  genipv  7508  genpelxp  7510  addcomprg  7577  mulcomprg  7579  uzval  9530  ixxval  9896  fzval  10010  hashinfom  10758  hashennn  10760  shftfn  10833  gcdval  11960  lcmval  12063  isprm  12109  odzval  12241  pceulem  12294  pceu  12295  pcval  12296  pczpre  12297  pcdiv  12302  istopon  13516  toponsspwpwg  13525  clsval  13614  neival  13646  cnpval  13701  blvalps  13891  blval  13892  limccl  14131  ellimc3apf  14132  eldvap  14154
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