Theorem cbvrabw 3453

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. Version of cbvrab 3456 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by Andrew Salmon, 11-Jul-2011.) Avoid ax-13 2406. (Revised by GG, 10-Jan-2024.) Avoid ax-10 2179. (Revised by Wolf Lammen, 19-Jul-2025.)

Hypotheses

Ref	Expression
cbvrabw.1	⊢ Ⅎ𝑥𝐴
cbvrabw.2	⊢ Ⅎ𝑦𝐴
cbvrabw.3	⊢ Ⅎ𝑦𝜑
cbvrabw.4	⊢ Ⅎ𝑥𝜓
cbvrabw.5	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvrabw	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑦)   𝐴(𝑥, 𝑦)

Proof of Theorem cbvrabw

Step	Hyp	Ref	Expression
1		cbvrabw.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
2	1	nfcri 2919	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
3		cbvrabw.3	. . . 4 ⊢ Ⅎ𝑦𝜑
4	2, 3	nfan 1932	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
5		cbvrabw.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
6	5	nfcri 2919	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
7		cbvrabw.4	. . . 4 ⊢ Ⅎ𝑥𝜓
8	6, 7	nfan 1932	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)
9		eleq1w 2848	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10		cbvrabw.5	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
11	9, 10	anbi12d 644	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
12	4, 8, 11	cbvabw 2836	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
13		df-rab 3419	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
14		df-rab 3419	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜓} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
15	12, 13, 14	3eqtr4i 2798	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146  {cab 2743  Ⅎwnfc 2912  {crab 3418

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 2148  ax-9 2156  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3419

This theorem is used by:  elrabsf  3791  f1ossf1o  7128  tfis  7857  cantnflem1  9665  scottexsOLD  9879  scott0bsOLD  9881  elmptrab  24035  bnj1534  35306  scottexf  38875  scott0f  38876  aks6d1c7lem3  43007  unitscyglem3  43022  unitscyglem4  43023  eq0rabdioph  43565  rexrabdioph  43579  rexfrabdioph  43580  elnn0rabdioph  43588  dvdsrabdioph  43595  binomcxplemdvsum  45123  fnlimcnv  46439  fnlimabslt  46451  stoweidlem34  46806  stoweidlem59  46831  pimltmnf2f  47469  pimgtpnf2f  47477  pimltpnf2f  47484  issmff  47506  smfpimltxrmptf  47530  smfpreimagtf  47540  smflim  47549  smfpimgtxr  47552  smfpimgtxrmptf  47556  smflim2  47578  smflimsup  47600  smfliminf  47603