Description: Version of axc10 with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 14-Jun-2019) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | bj-axc10v | |- ( A. x ( x = y -> A. x ph ) -> ph ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax6v | |- -. A. x -. x = y |
|
| 2 | con3 | |- ( ( x = y -> A. x ph ) -> ( -. A. x ph -> -. x = y ) ) |
|
| 3 | 2 | al2imi | |- ( A. x ( x = y -> A. x ph ) -> ( A. x -. A. x ph -> A. x -. x = y ) ) |
| 4 | 1 3 | mtoi | |- ( A. x ( x = y -> A. x ph ) -> -. A. x -. A. x ph ) |
| 5 | axc7 | |- ( -. A. x -. A. x ph -> ph ) |
|
| 6 | 4 5 | syl | |- ( A. x ( x = y -> A. x ph ) -> ph ) |