Theorem cbvabw 2293

Description: Version of cbvab 2294 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)

Hypotheses

Ref	Expression
cbvabw.1	⊢ Ⅎ𝑦𝜑
cbvabw.2	⊢ Ⅎ𝑥𝜓
cbvabw.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvabw	⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvabw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvabw.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
2	1	nfsbv 1940	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
3		equequ2 1706	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
4	3	imbi1d 230	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑧 → 𝜑)))
5	4	albidv 1817	. . . . . 6 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
6		sb6 1879	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
7		sb6 1879	. . . . . 6 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑))
8	5, 6, 7	3bitr4g 222	. . . . 5 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
9	2, 8	sbiev 1785	. . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
10		cbvabw.2	. . . . . 6 ⊢ Ⅎ𝑥𝜓
11		cbvabw.3	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
12	10, 11	sbiev 1785	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
13	12	sbbii 1758	. . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
14	9, 13	bitr3i 185	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
15		df-clab 2157	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
16		df-clab 2157	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
17	14, 15, 16	3bitr4i 211	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓})
18	17	eqriv 2167	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1346   = wceq 1348  Ⅎwnf 1453  [wsb 1755   ∈ wcel 2141  {cab 2156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163

This theorem is referenced by:  cbvsbcw  2982