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Theorem cbvabv 2161
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
Hypothesis
Ref Expression
cbvabv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvabv {𝑥𝜑} = {𝑦𝜓}
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvabv
StepHypRef Expression
1 nfv 1421 . 2 𝑦𝜑
2 nfv 1421 . 2 𝑥𝜓
3 cbvabv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvab 2160 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  {cab 2026
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
This theorem is referenced by:  cdeqab1  2756  difjust  2919  unjust  2921  injust  2923  uniiunlem  3028  dfif3  3343  pwjust  3360  snjust  3380  intab  3644  iotajust  4866  tfrlemi1  5946  frecsuc  5991  nqprlu  6643  recexpr  6734  caucvgprprlemval  6784  caucvgprprlemnbj  6789  caucvgprprlemaddq  6804  caucvgprprlem1  6805  caucvgprprlem2  6806  axcaucvg  6972  bds  9969
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