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Mirrors > Home > MPE Home > Th. List > cbvreuv | Structured version Visualization version GIF version |
Description: Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
cbvralv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvreuv | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1883 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1883 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvralv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvreu 3199 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∃!wreu 2943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-cleq 2644 df-clel 2647 df-reu 2948 |
This theorem is referenced by: reu8 3435 aceq1 8978 aceq2 8980 fin23lem27 9188 divalglem10 15172 lspsneu 19171 lshpsmreu 34714 wessf1ornlem 39685 fourierdlem50 40691 |
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