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Theorem cbvopabv 4648
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 1829 . 2 𝑧𝜑
2 nfv 1829 . 2 𝑤𝜑
3 nfv 1829 . 2 𝑥𝜓
4 nfv 1829 . 2 𝑦𝜓
5 cbvopabv.1 . 2 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
61, 2, 3, 4, 5cbvopab 4647 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  {copab 4636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638
This theorem is referenced by:  cantnf  8450  infxpen  8697  axdc2  9131  fpwwe2cbv  9308  fpwwecbv  9322  sylow1  17787  bcth  22851  vitali  23105  lgsquadlem3  24824  lgsquad  24825  islnopp  25349  ishpg  25369  hpgbr  25370  trgcopy  25414  trgcopyeu  25416  acopyeu  25443  tgasa1  25457  axcontlem1  25562  eulerpartlemgvv  29571  eulerpart  29577  cvmlift2lem13  30357  pellex  36213  aomclem8  36445
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