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Theorem cbvopabv 5129
Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)
Hypothesis
Ref Expression
cbvopabv.1 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbvopabv {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤   𝜑,𝑧,𝑤   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑧,𝑤)

Proof of Theorem cbvopabv
StepHypRef Expression
1 nfv 1906 . 2 𝑧𝜑
2 nfv 1906 . 2 𝑤𝜑
3 nfv 1906 . 2 𝑥𝜓
4 nfv 1906 . 2 𝑦𝜓
5 cbvopabv.1 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
61, 2, 3, 4, 5cbvopab 5128 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  {copab 5119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120
This theorem is referenced by:  cantnf  9144  infxpen  9428  axdc2  9859  fpwwe2cbv  10040  fpwwecbv  10054  sylow1  18657  bcth  23859  vitali  24141  lgsquadlem3  25885  lgsquad  25886  islnopp  26452  ishpg  26472  hpgbr  26473  trgcopy  26517  trgcopyeu  26519  acopyeu  26547  tgasa1  26571  axcontlem1  26677  eulerpartlemgvv  31533  eulerpart  31539  cvmlift2lem13  32459  pellex  39310  aomclem8  39539  sprsymrelf  43534
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