Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvopabv Structured version   Visualization version   GIF version

Theorem cbvopabv 4755
 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 1883 . 2 𝑧𝜑
2 nfv 1883 . 2 𝑤𝜑
3 nfv 1883 . 2 𝑥𝜓
4 nfv 1883 . 2 𝑦𝜓
5 cbvopabv.1 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
61, 2, 3, 4, 5cbvopab 4754 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  {copab 4745 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746 This theorem is referenced by:  cantnf  8628  infxpen  8875  axdc2  9309  fpwwe2cbv  9490  fpwwecbv  9504  sylow1  18064  bcth  23172  vitali  23427  lgsquadlem3  25152  lgsquad  25153  islnopp  25676  ishpg  25696  hpgbr  25697  trgcopy  25741  trgcopyeu  25743  acopyeu  25770  tgasa1  25784  axcontlem1  25889  eulerpartlemgvv  30566  eulerpart  30572  cvmlift2lem13  31423  pellex  37716  aomclem8  37948  sprsymrelf  42070
 Copyright terms: Public domain W3C validator