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Mirrors > Home > MPE Home > Th. List > cbvrexsvw | Structured version Visualization version GIF version |
Description: Change bound variable by using a substitution. Version of cbvrexsv 3365 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by NM, 2-Mar-2008.) Avoid ax-13 2375. (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 1912 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfs1v 2154 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
3 | sbequ12 2249 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvrexw 3305 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 [wsb 2062 ∃wrex 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-10 2139 ax-11 2155 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 df-sb 2063 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 |
This theorem is referenced by: rspesbca 3890 ac6sf 10527 ac6gf 37719 |
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