Description: Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | ovanraleqv.1 | |- ( B = X -> ( ph <-> ps ) ) |
|
| Assertion | ovanraleqv | |- ( B = X -> ( A. x e. V ( ph /\ ( A .x. B ) = C ) <-> A. x e. V ( ps /\ ( A .x. X ) = C ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ovanraleqv.1 | |- ( B = X -> ( ph <-> ps ) ) |
|
| 2 | oveq2 | |- ( B = X -> ( A .x. B ) = ( A .x. X ) ) |
|
| 3 | 2 | eqeq1d | |- ( B = X -> ( ( A .x. B ) = C <-> ( A .x. X ) = C ) ) |
| 4 | 1 3 | anbi12d | |- ( B = X -> ( ( ph /\ ( A .x. B ) = C ) <-> ( ps /\ ( A .x. X ) = C ) ) ) |
| 5 | 4 | ralbidv | |- ( B = X -> ( A. x e. V ( ph /\ ( A .x. B ) = C ) <-> A. x e. V ( ps /\ ( A .x. X ) = C ) ) ) |