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Theorem cbvrabv 2556
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 2178 . 2 𝑥𝐴
2 nfcv 2178 . 2 𝑦𝐴
3 nfv 1421 . 2 𝑦𝜑
4 nfv 1421 . 2 𝑥𝜓
5 cbvrabv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
61, 2, 3, 4, 5cbvrab 2555 1 {𝑥𝐴𝜑} = {𝑦𝐴𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  {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-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315
This theorem is referenced by:  pwnss  3912  acexmidlemv  5510  genipv  6605  ltexpri  6709
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