Theorem cbvrab 2858

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. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem cbvrab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . 4 ⊢ Ⅎz(x ∈ A ∧ φ)
2		cbvrab.1	. . . . . 6 ⊢ ℲxA
3	2	nfcri 2484	. . . . 5 ⊢ Ⅎx z ∈ A
4		nfs1v 2106	. . . . 5 ⊢ Ⅎx[z / x]φ
5	3, 4	nfan 1824	. . . 4 ⊢ Ⅎx(z ∈ A ∧ [z / x]φ)
6		eleq1 2413	. . . . 5 ⊢ (x = z → (x ∈ A ↔ z ∈ A))
7		sbequ12 1919	. . . . 5 ⊢ (x = z → (φ ↔ [z / x]φ))
8	6, 7	anbi12d 691	. . . 4 ⊢ (x = z → ((x ∈ A ∧ φ) ↔ (z ∈ A ∧ [z / x]φ)))
9	1, 5, 8	cbvab 2472	. . 3 ⊢ {x ∣ (x ∈ A ∧ φ)} = {z ∣ (z ∈ A ∧ [z / x]φ)}
10		cbvrab.2	. . . . . 6 ⊢ ℲyA
11	10	nfcri 2484	. . . . 5 ⊢ Ⅎy z ∈ A
12		cbvrab.3	. . . . . 6 ⊢ Ⅎyφ
13	12	nfsb 2109	. . . . 5 ⊢ Ⅎy[z / x]φ
14	11, 13	nfan 1824	. . . 4 ⊢ Ⅎy(z ∈ A ∧ [z / x]φ)
15		nfv 1619	. . . 4 ⊢ Ⅎz(y ∈ A ∧ ψ)
16		eleq1 2413	. . . . 5 ⊢ (z = y → (z ∈ A ↔ y ∈ A))
17		sbequ 2060	. . . . . 6 ⊢ (z = y → ([z / x]φ ↔ [y / x]φ))
18		cbvrab.4	. . . . . . 7 ⊢ Ⅎxψ
19		cbvrab.5	. . . . . . 7 ⊢ (x = y → (φ ↔ ψ))
20	18, 19	sbie 2038	. . . . . 6 ⊢ ([y / x]φ ↔ ψ)
21	17, 20	syl6bb 252	. . . . 5 ⊢ (z = y → ([z / x]φ ↔ ψ))
22	16, 21	anbi12d 691	. . . 4 ⊢ (z = y → ((z ∈ A ∧ [z / x]φ) ↔ (y ∈ A ∧ ψ)))
23	14, 15, 22	cbvab 2472	. . 3 ⊢ {z ∣ (z ∈ A ∧ [z / x]φ)} = {y ∣ (y ∈ A ∧ ψ)}
24	9, 23	eqtri 2373	. 2 ⊢ {x ∣ (x ∈ A ∧ φ)} = {y ∣ (y ∈ A ∧ ψ)}
25		df-rab 2624	. 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
26		df-rab 2624	. 2 ⊢ {y ∈ A ∣ ψ} = {y ∣ (y ∈ A ∧ ψ)}
27	24, 25, 26	3eqtr4i 2383	1 ⊢ {x ∈ A ∣ φ} = {y ∈ A ∣ ψ}

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  {crab 2619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624

This theorem is referenced by:  cbvrabv  2859  elrabsf  3085