Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 4-Oct-2016) (Revised by BJ, 17-Jun-2019) Avoid ax-10 . (Revised by Gino Giotto, 18-Nov-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | dral1v.1 | |- ( A. x x = y -> ( ph <-> ps ) ) |
|
| Assertion | drnf1v | |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dral1v.1 | |- ( A. x x = y -> ( ph <-> ps ) ) |
|
| 2 | 1 | drex1v | |- ( A. x x = y -> ( E. x ph <-> E. y ps ) ) |
| 3 | 1 | dral1v | |- ( A. x x = y -> ( A. x ph <-> A. y ps ) ) |
| 4 | 2 3 | imbi12d | |- ( A. x x = y -> ( ( E. x ph -> A. x ph ) <-> ( E. y ps -> A. y ps ) ) ) |
| 5 | df-nf | |- ( F/ x ph <-> ( E. x ph -> A. x ph ) ) |
|
| 6 | df-nf | |- ( F/ y ps <-> ( E. y ps -> A. y ps ) ) |
|
| 7 | 4 5 6 | 3bitr4g | |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) ) |