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Theorem elrabd 2897
Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2895. (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 306 . 2 (𝜑 → (𝐴𝐵𝜒))
4 elrabd.1 . . 3 (𝑥 = 𝐴 → (𝜓𝜒))
54elrab 2895 . 2 (𝐴 ∈ {𝑥𝐵𝜓} ↔ (𝐴𝐵𝜒))
63, 5sylibr 134 1 (𝜑𝐴 ∈ {𝑥𝐵𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  {crab 2459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741
This theorem is referenced by:  ctssdccl  7112  suplocexprlemru  7720  suplocexprlemloc  7722  zsupssdc  11957  uzwodc  12040  phisum  12242  odzcllem  12244  pcpremul  12295  znnen  12401  ennnfonelemj0  12404  ennnfonelemg  12406  issubmd  12870  mhmeql  12881  cdivcncfap  14126  cnopnap  14133  ivthinc  14160  limcdifap  14170  limcimolemlt  14172  dvcoapbr  14210  subctctexmid  14789
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