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Theorem cbvabv 2208
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 1464 . 2 𝑦𝜑
2 nfv 1464 . 2 𝑥𝜓
3 cbvabv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvab 2207 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1287  {cab 2071
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078
This theorem is referenced by:  cdeqab1  2821  difjust  2989  unjust  2991  injust  2993  uniiunlem  3098  dfif3  3392  pwjust  3416  snjust  3436  intab  3700  iotajust  4942  tfrlemi1  6045  tfr1onlemaccex  6061  tfrcllemaccex  6074  frecsuc  6120  isbth  6613  nqprlu  7043  recexpr  7134  caucvgprprlemval  7184  caucvgprprlemnbj  7189  caucvgprprlemaddq  7204  caucvgprprlem1  7205  caucvgprprlem2  7206  axcaucvg  7372  bds  11172
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