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Theorem cbvopabv 5105
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 1915 . 2 𝑧𝜑
2 nfv 1915 . 2 𝑤𝜑
3 nfv 1915 . 2 𝑥𝜓
4 nfv 1915 . 2 𝑦𝜓
5 cbvopabv.1 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
61, 2, 3, 4, 5cbvopab 5104 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  {copab 5095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-un 3889  df-sn 4529  df-pr 4531  df-op 4535  df-opab 5096
This theorem is referenced by:  cantnf  9144  infxpen  9429  axdc2  9864  fpwwe2cbv  10045  fpwwecbv  10059  sylow1  18723  bcth  23936  vitali  24220  lgsquadlem3  25969  lgsquad  25970  islnopp  26536  ishpg  26556  hpgbr  26557  trgcopy  26601  trgcopyeu  26603  acopyeu  26631  tgasa1  26655  axcontlem1  26761  eulerpartlemgvv  31742  eulerpart  31748  cvmlift2lem13  32670  pellex  39763  aomclem8  39992  sprsymrelf  43999
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