Theorem cbvopabv 5178

Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.) Reduce axiom usage. (Revised by GG, 15-Oct-2024.)

Hypothesis

Ref	Expression
cbvopabv.1	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))

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

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq12 4835	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
2	1	eqeq2d 2772	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑧, 𝑤⟩))
3		cbvopabv.1	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
4	2, 3	anbi12d 644	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)))
5	4	cbvex2vw 2074	. . 3 ⊢ (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓))
6	5	abbii 2828	. 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
7		df-opab 5168	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
8		df-opab 5168	. 2 ⊢ {⟨𝑧, 𝑤⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
9	6, 7, 8	3eqtr4i 2794	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812  {cab 2739  ⟨cop 4590  {copab 5167

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168

This theorem is used by:  cantnf  9694  infxpen  10093  axdc2  10527  fpwwe2cbv  10715  fpwwecbv  10729  sylow1  19817  bcth  25650  vitali  25934  lgsquadlem3  27709  lgsquad  27710  islnopp  29215  ishpg  29237  hpgbr  29238  elplngid  29260  lnincplng  29262  plngcp  29264  plngrot  29268  nhpmirhp  29276  lnperpexs  29310  trgcopy  29311  trgcopyeu  29313  acopyeu  29342  ragraghl  29346  tgaaddcpbllem2  29350  tgaaddcpbl2  29353  angmgmaddeu1  29379  tgasa1  29403  prlnghpg  29424  prlngmo  29432  axcontlem1  29542  constrext2chn  34391  eulerpartlemgvv  35008  eulerpart  35014  cvmlift2lem13  36080  pellex  43841  aomclem8  44062  sprsymrelf  48576