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Theorem cbvopabv 4654
 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 1830 . 2 𝑧𝜑
2 nfv 1830 . 2 𝑤𝜑
3 nfv 1830 . 2 𝑥𝜓
4 nfv 1830 . 2 𝑦𝜓
5 cbvopabv.1 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
61, 2, 3, 4, 5cbvopab 4653 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  {copab 4642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644 This theorem is referenced by:  cantnf  8473  infxpen  8720  axdc2  9154  fpwwe2cbv  9331  fpwwecbv  9345  sylow1  17841  bcth  22934  vitali  23188  lgsquadlem3  24907  lgsquad  24908  islnopp  25431  ishpg  25451  hpgbr  25452  trgcopy  25496  trgcopyeu  25498  acopyeu  25525  tgasa1  25539  axcontlem1  25644  eulerpartlemgvv  29765  eulerpart  29771  cvmlift2lem13  30551  pellex  36417  aomclem8  36649
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