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Theorem elrabd 2866
 Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2864. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
elrabd.1 (𝑥 = 𝐴 → (𝜓𝜒))
elrabd.2 (𝜑𝐴𝐵)
elrabd.3 (𝜑𝜒)
Assertion
Ref Expression
elrabd (𝜑𝐴 ∈ {𝑥𝐵𝜓})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem elrabd
StepHypRef Expression
1 elrabd.2 . . 3 (𝜑𝐴𝐵)
2 elrabd.3 . . 3 (𝜑𝜒)
31, 2jca 304 . 2 (𝜑 → (𝐴𝐵𝜒))
4 elrabd.1 . . 3 (𝑥 = 𝐴 → (𝜓𝜒))
54elrab 2864 . 2 (𝐴 ∈ {𝑥𝐵𝜓} ↔ (𝐴𝐵𝜒))
63, 5sylibr 133 1 (𝜑𝐴 ∈ {𝑥𝐵𝜓})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 2125  {crab 2436 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rab 2441  df-v 2711 This theorem is referenced by:  ctssdccl  7041  suplocexprlemru  7618  suplocexprlemloc  7620  znnen  12078  ennnfonelemj0  12081  ennnfonelemg  12083  cdivcncfap  12926  cnopnap  12933  ivthinc  12960  limcdifap  12970  limcimolemlt  12972  dvcoapbr  13010  subctctexmid  13512
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