Theorem cbvrab 1425

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound variable hypotheses in place of distinct variable conditions.

Hypotheses

Ref	Expression
cbvrab.1	⊢ (z ∈ A → ∀x z ∈ A)
cbvrab.2	⊢ (z ∈ A → ∀y z ∈ A)
cbvrab.3	⊢ (φ → ∀yφ)
cbvrab.4	⊢ (ψ → ∀xψ)
cbvrab.5	⊢ (x = y → (φ ↔ ψ))

Assertion

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

Distinct variable group(s):   x,y,z   z,A

Proof of Theorem cbvrab

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 ⊢ (z ∈ x → ∀y z ∈ x)
2		cbvrab.2	. . . . 5 ⊢ (z ∈ A → ∀y z ∈ A)
3	1, 2	hbel 1172	. . . 4 ⊢ (x ∈ A → ∀y x ∈ A)
4		cbvrab.3	. . . 4 ⊢ (φ → ∀yφ)
5	3, 4	hban 704	. . 3 ⊢ ((x ∈ A ∧ φ) → ∀y(x ∈ A ∧ φ))
6		ax-17 925	. . . . 5 ⊢ (z ∈ y → ∀x z ∈ y)
7		cbvrab.1	. . . . 5 ⊢ (zb/FONT> ∈ A → ∀x z ∈ A)
8	6, 7	hbel 1172	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
9		cbvrab.4	. . . 4 ⊢ (ψ → ∀xψ)
10	8, 9	hban 704	. . 3 ⊢ ((y ∈ A ∧ ψ) → ∀x(y ∈ A ∧ ψ))
11		eleq1 1149	. . . 4 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
12		cbvrab.5	. . . 4 ⊢ (x = y → (φ ↔ ψ))
13	11, 12	anbi12d 476	. . 3 ⊢ (x = y → ((x ∈ A ∧ φ) ↔ (y ∈ A ∧ ψ)))
14	5, 10, 13	cbvab 1423	. 2 ⊢ {x∣(x ∈ A ∧ φ)} = {y∣(y ∈ A ∧ ψ)}
15		df-rab 1208	. 2 ⊢ {x ∈ A∣φ} = {x∣(x ∈ A ∧ φ)}
16		df-rab 1208	. 2 ⊢ {y ∈ A∣ψ} = {y∣(y ∈ A ∧ ψ)}
17	14, 15, 16	3eqtr4 1126	1 ⊢ {x ∈ A∣φ} = {y ∈ A∣ψ}

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  {crab 1204

This theorem is referenced by:  cbvrabv 1426  elrabsf 1456  iunrab 2022  scottexs 3543  scott0s 3544  hta 3619

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208  df-v 1349

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