Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cbvreuvw | Structured version Visualization version GIF version |
Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv 3454 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 5-Apr-2004.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbvralvw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvreuvw | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1915 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1915 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvralvw.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvreuw 3443 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∃!wreu 3140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-10 2145 ax-11 2161 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clel 2893 df-reu 3145 |
This theorem is referenced by: reu8 3724 aceq1 9543 aceq2 9545 fin23lem27 9750 divalglem10 15753 lspsneu 19895 lshpsmreu 36260 wessf1ornlem 41494 fourierdlem50 42490 |
Copyright terms: Public domain | W3C validator |