Theorem cbvrabw 3446

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. Version of cbvrab 3449 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by Andrew Salmon, 11-Jul-2011.) Avoid ax-13 2401. (Revised by GG, 10-Jan-2024.) Avoid ax-10 2177. (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 2914	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
3		cbvrabw.3	. . . 4 ⊢ Ⅎ𝑦𝜑
4	2, 3	nfan 1931	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
5		cbvrabw.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
6	5	nfcri 2914	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
7		cbvrabw.4	. . . 4 ⊢ Ⅎ𝑥𝜓
8	6, 7	nfan 1931	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)
9		eleq1w 2843	. . . 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 2831	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
13		df-rab 3413	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
14		df-rab 3413	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜓} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
15	12, 13, 14	3eqtr4i 2793	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144  {cab 2738  Ⅎwnfc 2907  {crab 3412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rab 3413

This theorem is used by:  elrabsf  3783  f1ossf1o  7132  tfis  7868  cantnflem1  9695  scottexsOLD  9948  scott0bsOLD  9950  elmptrab  24153  bnj1534  35531  rankse  35795  scottexf  39207  scott0f  39208  aks6d1c7lem3  43339  unitscyglem3  43354  unitscyglem4  43355  eq0rabdioph  43894  rexrabdioph  43908  rexfrabdioph  43909  elnn0rabdioph  43917  dvdsrabdioph  43924  binomcxplemdvsum  45446  fnlimcnv  46784  fnlimabslt  46796  stoweidlem34  47151  stoweidlem59  47176  pimltmnf2f  47814  pimgtpnf2f  47822  pimltpnf2f  47829  issmff  47851  smfpimltxrmptf  47875  smfpreimagtf  47885  smflim  47894  smfpimgtxr  47897  smfpimgtxrmptf  47901  smflim2  47923  smflimsup  47945  smfliminf  47948