Theorem cbvrabcsfw 3938

Description: Version of cbvrabcsf 3942 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Andrew Salmon, 13-Jul-2011.) (Revised by Gino Giotto, 26-Jan-2024.)

Hypotheses

Ref	Expression
cbvrabcsfw.1	⊢ Ⅎ𝑦𝐴
cbvrabcsfw.2	⊢ Ⅎ𝑥𝐵
cbvrabcsfw.3	⊢ Ⅎ𝑦𝜑
cbvrabcsfw.4	⊢ Ⅎ𝑥𝜓
cbvrabcsfw.5	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
cbvrabcsfw.6	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvrabcsfw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1916	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑)
2		nfcsb1v 3919	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴
3	2	nfcri 2889	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴
4		nfs1v 2152	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
5	3, 4	nfan 1901	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑)
6		id 22	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
7		csbeq1a 3908	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
8	6, 7	eleq12d 2826	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴))
9		sbequ12 2242	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
10	8, 9	anbi12d 630	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑)))
11	1, 5, 10	cbvabw 2805	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑)}
12		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑦𝑧
13		cbvrabcsfw.1	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
14	12, 13	nfcsbw 3921	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑧 / 𝑥⦌𝐴
15	14	nfcri 2889	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴
16		cbvrabcsfw.3	. . . . . 6 ⊢ Ⅎ𝑦𝜑
17	16	nfsbv 2322	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
18	15, 17	nfan 1901	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑)
19		nfv 1916	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐵 ∧ 𝜓)
20		id 22	. . . . . 6 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
21		csbeq1 3897	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
22		vex 3477	. . . . . . . 8 ⊢ 𝑦 ∈ V
23		cbvrabcsfw.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
24		cbvrabcsfw.5	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
25	22, 23, 24	csbief 3929	. . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = 𝐵
26	21, 25	eqtrdi 2787	. . . . . 6 ⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐴 = 𝐵)
27	20, 26	eleq12d 2826	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ↔ 𝑦 ∈ 𝐵))
28		cbvrabcsfw.4	. . . . . 6 ⊢ Ⅎ𝑥𝜓
29		cbvrabcsfw.6	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
30	28, 29	sbhypf 3538	. . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
31	27, 30	anbi12d 630	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐵 ∧ 𝜓)))
32	18, 19, 31	cbvabw 2805	. . 3 ⊢ {𝑧 ∣ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜓)}
33	11, 32	eqtri 2759	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜓)}
34		df-rab 3432	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
35		df-rab 3432	. 2 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜓} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜓)}
36	33, 34, 35	3eqtr4i 2769	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540  Ⅎwnf 1784  [wsb 2066   ∈ wcel 2105  {cab 2708  Ⅎwnfc 2882  {crab 3431  ⦋csb 3894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895

This theorem is referenced by:  cbvrabv2w  44120  smfsup  45830  smfinflem  45833  smfinf  45834