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Theorem rabbidv 2549
 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 261 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32rabbidva 2548 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {crab 2310 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-ral 2311  df-rab 2315 This theorem is referenced by:  rabeqbidv  2552  difeq2  3056  seex  4072  mptiniseg  4815  cardcl  6361  isnumi  6362  cardval3ex  6365  carden2bex  6369  genpdflem  6605  genipv  6607  genpelxp  6609  addcomprg  6676  mulcomprg  6678  uzval  8475  ixxval  8765  fzval  8876  shftfn  9425
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