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| Description: Change bound variable by using a substitution. Version of cbvralsv 3366 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.) Avoid ax-10 2141, ax-12 2177. (Revised by SN, 21-Aug-2025.) | 
| Ref | Expression | 
|---|---|
| cbvralsvw | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sb8v 2355 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦[𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝜑)) | |
| 2 | df-ral 3062 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
| 3 | df-ral 3062 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑)) | |
| 4 | eleq1w 2824 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
| 5 | 4 | imbi1d 341 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜑))) | 
| 6 | 5 | sbbiiev 2092 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝜑) ↔ [𝑦 / 𝑥](𝑦 ∈ 𝐴 → 𝜑)) | 
| 7 | sbrimvw 2091 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝑦 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑)) | |
| 8 | 6, 7 | bitr2i 276 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑) ↔ [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝜑)) | 
| 9 | 8 | albii 1819 | . . 3 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦[𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝜑)) | 
| 10 | 3, 9 | bitri 275 | . 2 ⊢ (∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦[𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝜑)) | 
| 11 | 1, 2, 10 | 3bitr4i 303 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 [wsb 2064 ∈ wcel 2108 ∀wral 3061 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-11 2157 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clel 2816 df-ral 3062 | 
| This theorem is referenced by: sbralieALT 3359 rspsbc 3879 ralxpf 5857 tfinds 7881 tfindes 7884 nn0min 32822 | 
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