Description: Alternate proof of mobidv directly from its analogues albidv and exbidv , using deduction style. Note the proof structure, similar to mobi . (Contributed by Mario Carneiro, 7-Oct-2016) Reduce axiom dependencies and shorten proof. Remove dependency on ax-12 by adapting proof of mobid . (Revised by BJ, 26-Sep-2022) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | mobidvALT.1 | |- ( ph -> ( ps <-> ch ) ) |
|
| Assertion | mobidvALT | |- ( ph -> ( E* x ps <-> E* x ch ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mobidvALT.1 | |- ( ph -> ( ps <-> ch ) ) |
|
| 2 | 1 | imbi1d | |- ( ph -> ( ( ps -> x = y ) <-> ( ch -> x = y ) ) ) |
| 3 | 2 | albidv | |- ( ph -> ( A. x ( ps -> x = y ) <-> A. x ( ch -> x = y ) ) ) |
| 4 | 3 | exbidv | |- ( ph -> ( E. y A. x ( ps -> x = y ) <-> E. y A. x ( ch -> x = y ) ) ) |
| 5 | dfmo | |- ( E* x ps <-> E. y A. x ( ps -> x = y ) ) |
|
| 6 | dfmo | |- ( E* x ch <-> E. y A. x ( ch -> x = y ) ) |
|
| 7 | 4 5 6 | 3bitr4g | |- ( ph -> ( E* x ps <-> E* x ch ) ) |