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| Description: Change bound variable by using a substitution. Version of cbvrexsv 3367 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 2-Mar-2008.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.) | 
| Ref | Expression | 
|---|---|
| cbvrexsvw | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfv 1914 | . 2 ⊢ Ⅎ𝑦𝜑 | |
| 2 | nfs1v 2156 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
| 3 | sbequ12 2251 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
| 4 | 1, 2, 3 | cbvrexw 3307 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 [wsb 2064 ∃wrex 3070 | 
| 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-10 2141 ax-11 2157 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 | 
| This theorem is referenced by: rspesbca 3881 ac6sf 10529 ac6gf 37739 | 
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