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Theorem cbvral 2627
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
Hypotheses
Ref Expression
cbvral.1 𝑦𝜑
cbvral.2 𝑥𝜓
cbvral.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvral (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvral
StepHypRef Expression
1 nfcv 2258 . 2 𝑥𝐴
2 nfcv 2258 . 2 𝑦𝐴
3 cbvral.1 . 2 𝑦𝜑
4 cbvral.2 . 2 𝑥𝜓
5 cbvral.3 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
61, 2, 3, 4, 5cbvralf 2625 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wnf 1421  wral 2393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398
This theorem is referenced by:  cbvralv  2631  cbvralsv  2642  cbviin  3821  frind  4244  ralxpf  4655  eqfnfv2f  5490  ralrnmpt  5530  dff13f  5639  ofrfval2  5966  fmpox  6066  cbvixp  6577  mptelixpg  6596  xpf1o  6706  indstr  9356  fsum3  11124  fsum00  11199  mertenslem2  11273  cnmpt11  12379  cnmpt21  12387  bj-bdfindes  13074  bj-findes  13106
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