Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmov with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 16-Jun-2017) (Revised by Gino Giotto, 30-Sep-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | cbvralvw.1 | |- ( x = y -> ( ph <-> ps ) ) |
|
| Assertion | cbvrmovw | |- ( 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 | cbvmovw | |- ( E* x ( x e. A /\ ph ) <-> E* y ( y e. A /\ ps ) ) |
| 5 | df-rmo | |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) ) |
|
| 6 | df-rmo | |- ( 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 ) |