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Theorem elrab3 2699
 Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
Hypothesis
Ref Expression
elrab.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elrab3 (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elrab3
StepHypRef Expression
1 elrab.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
21elrab 2698 . 2 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
32baib 828 1 (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
 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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  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-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559 This theorem is referenced by:  unimax  3614  frind  4089  ordtriexmidlem2  4246  ordtriexmid  4247  ordtri2orexmid  4248  onsucelsucexmid  4255  0elsucexmid  4289  ordpwsucexmid  4294  ordtri2or2exmid  4296  acexmidlema  5503  acexmidlemb  5504  isnumi  6362  genpelvl  6610  genpelvu  6611  cauappcvgprlemladdru  6754  cauappcvgprlem1  6757  caucvgprlem1  6777  ublbneg  8548  negm  8550
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