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Theorem elrabd 2896
Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2894. (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 2894 . 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 2740
This theorem is referenced by:  ctssdccl  7110  suplocexprlemru  7718  suplocexprlemloc  7720  zsupssdc  11955  uzwodc  12038  phisum  12240  odzcllem  12242  pcpremul  12293  znnen  12399  ennnfonelemj0  12402  ennnfonelemg  12404  issubmd  12865  mhmeql  12876  cdivcncfap  14090  cnopnap  14097  ivthinc  14124  limcdifap  14134  limcimolemlt  14136  dvcoapbr  14174  subctctexmid  14753
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