Theorem cbvrab 3449

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. Usage of this theorem is discouraged because it depends on ax-13 2401. Use the weaker cbvrabw 3446 when possible. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)

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 1946	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑)
2		cbvrab.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2914	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
4		nfs1v 2192	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
5	3, 4	nfan 1931	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
6		eleq1w 2843	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
7		sbequ12 2286	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
8	6, 7	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)))
9	1, 5, 8	cbvab 2832	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)}
10		cbvrab.2	. . . . . 6 ⊢ Ⅎ𝑦𝐴
11	10	nfcri 2914	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
12		cbvrab.3	. . . . . 6 ⊢ Ⅎ𝑦𝜑
13	12	nfsb 2552	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
14	11, 13	nfan 1931	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
15		nfv 1946	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓)
16		eleq1w 2843	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17		sbequ 2119	. . . . . 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 2531	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
21	17, 20	bitrdi 290	. . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
22	16, 21	anbi12d 644	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
23	14, 15, 22	cbvab 2832	. . 3 ⊢ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
24	9, 23	eqtri 2783	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
25		df-rab 3413	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
26		df-rab 3413	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜓} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
27	24, 25, 26	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  [wsb 2098   ∈ 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-10 2177  ax-11 2193  ax-12 2212  ax-13 2401  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: (None)