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Theorem cbvralv 2533
 Description: Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)
Hypothesis
Ref Expression
cbvralv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvralv (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvralv
StepHypRef Expression
1 nfv 1421 . 2 𝑦𝜑
2 nfv 1421 . 2 𝑥𝜓
3 cbvralv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvral 2529 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wral 2306 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-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311 This theorem is referenced by:  cbvral2v  2541  cbvral3v  2543  reu7  2736  reusv3i  4191  cnvpom  4860  f1mpt  5410  grprinvlem  5695  grprinvd  5696  tfrlem1  5923  tfrlemiubacc  5944  tfrlemi1  5946  rdgon  5973  nneneq  6320  cauappcvgprlemladdrl  6755  caucvgprlemcl  6774  caucvgprlemladdrl  6776  caucvgsrlembound  6878  caucvgsrlemgt1  6879  caucvgsrlemoffres  6884  peano5nnnn  6966  axcaucvglemres  6973  nnsub  7952  ublbneg  8548  iseqovex  9219  iseqval  9220  monoord2  9236  caucvgre  9580  cvg1nlemcau  9583  resqrexlemglsq  9620  resqrexlemsqa  9622  resqrexlemex  9623  cau3lem  9710  sqrt2irr  9878
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