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Theorem cbvopabv 4856
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 1995 . 2 𝑧𝜑
2 nfv 1995 . 2 𝑤𝜑
3 nfv 1995 . 2 𝑥𝜓
4 nfv 1995 . 2 𝑦𝜓
5 cbvopabv.1 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
61, 2, 3, 4, 5cbvopab 4855 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  {copab 4846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-opab 4847
This theorem is referenced by:  cantnf  8754  infxpen  9037  axdc2  9473  fpwwe2cbv  9654  fpwwecbv  9668  sylow1  18225  bcth  23345  vitali  23601  lgsquadlem3  25328  lgsquad  25329  islnopp  25852  ishpg  25872  hpgbr  25873  trgcopy  25917  trgcopyeu  25919  acopyeu  25946  tgasa1  25960  axcontlem1  26065  eulerpartlemgvv  30778  eulerpart  30784  cvmlift2lem13  31635  pellex  37925  aomclem8  38157  sprsymrelf  42273
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