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Theorem cbvrabv 2680
Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)
Hypothesis
Ref Expression
cbvrabv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrabv {𝑥𝐴𝜑} = {𝑦𝐴𝜓}
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvrabv
StepHypRef Expression
1 nfcv 2279 . 2 𝑥𝐴
2 nfcv 2279 . 2 𝑦𝐴
3 nfv 1508 . 2 𝑦𝜑
4 nfv 1508 . 2 𝑥𝜓
5 cbvrabv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
61, 2, 3, 4, 5cbvrab 2679 1 {𝑥𝐴𝜑} = {𝑦𝐴𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  {crab 2418
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423
This theorem is referenced by:  pwnss  4078  acexmidlemv  5765  exmidac  7058  genipv  7310  ltexpri  7414  suplocsrlempr  7608  suplocsr  7610  sqne2sq  11844
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