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Theorem cbvral 2711
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 2329 . 2 𝑥𝐴
2 nfcv 2329 . 2 𝑦𝐴
3 cbvral.1 . 2 𝑦𝜑
4 cbvral.2 . 2 𝑥𝜓
5 cbvral.3 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
61, 2, 3, 4, 5cbvralf 2707 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wnf 1470  wral 2465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470
This theorem is referenced by:  cbvralv  2715  cbvralsv  2731  cbviin  3936  frind  4364  ralxpf  4785  eqfnfv2f  5630  ralrnmpt  5671  dff13f  5784  ofrfval2  6112  fmpox  6214  cbvixp  6728  mptelixpg  6747  xpf1o  6857  indstr  9606  fsum3  11408  fsum00  11483  mertenslem2  11557  fprodcl2lem  11626  fprodle  11661  ctiunctal  12455  cnmpt11  14054  cnmpt21  14062  bj-bdfindes  14972  bj-findes  15004
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