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Theorem rabbidv 2604
Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (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 270 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32rabbidva 2603 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1287  wcel 1436  {crab 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-ral 2360  df-rab 2364
This theorem is referenced by:  rabeqbidv  2610  difeq2  3101  seex  4136  mptiniseg  4891  supeq1  6625  supeq2  6628  supeq3  6629  cardcl  6753  isnumi  6754  cardval3ex  6757  carden2bex  6761  genpdflem  7010  genipv  7012  genpelxp  7014  addcomprg  7081  mulcomprg  7083  uzval  8953  ixxval  9246  fzval  9358  hashinfom  10083  hashennn  10085  shftfn  10155  gcdval  10833  lcmval  10927  isprm  10973
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