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Theorem rabbidv 2566
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 265 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32rabbidva 2565 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-ral 2328  df-rab 2332
This theorem is referenced by:  rabeqbidv  2569  difeq2  3083  seex  4099  mptiniseg  4842  supeq1  6391  supeq2  6394  supeq3  6395  cardcl  6418  isnumi  6419  cardval3ex  6422  carden2bex  6426  genpdflem  6662  genipv  6664  genpelxp  6666  addcomprg  6733  mulcomprg  6735  uzval  8570  ixxval  8865  fzval  8977  shftfn  9652
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