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Theorem cbvabv 2203
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 1462 . 2 𝑦𝜑
2 nfv 1462 . 2 𝑥𝜓
3 cbvabv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvab 2202 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  {cab 2068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075
This theorem is referenced by:  cdeqab1  2808  difjust  2975  unjust  2977  injust  2979  uniiunlem  3083  dfif3  3372  pwjust  3391  snjust  3411  intab  3673  iotajust  4896  tfrlemi1  5981  tfr1onlemaccex  5997  tfrcllemaccex  6010  frecsuc  6056  nqprlu  6799  recexpr  6890  caucvgprprlemval  6940  caucvgprprlemnbj  6945  caucvgprprlemaddq  6960  caucvgprprlem1  6961  caucvgprprlem2  6962  axcaucvg  7128  bds  10800
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