Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 5-Apr-2004) (Revised by Gino Giotto, 30-Sep-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | cbvralvw.1 | |- ( x = y -> ( ph <-> ps ) ) |
|
| Assertion | cbvreuvw | |- ( E! x e. A ph <-> E! y e. A ps ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvralvw.1 | |- ( x = y -> ( ph <-> ps ) ) |
|
| 2 | eleq1w | |- ( x = y -> ( x e. A <-> y e. A ) ) |
|
| 3 | 2 1 | anbi12d | |- ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. A /\ ps ) ) ) |
| 4 | 3 | cbveuvw | |- ( E! x ( x e. A /\ ph ) <-> E! y ( y e. A /\ ps ) ) |
| 5 | df-reu | |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) ) |
|
| 6 | df-reu | |- ( E! y e. A ps <-> E! y ( y e. A /\ ps ) ) |
|
| 7 | 4 5 6 | 3bitr4i | |- ( E! x e. A ph <-> E! y e. A ps ) |