Description: Alternate proof of ax13 from FOL, sp , and axc9 . (Contributed by NM, 21-Dec-2015) (Proof shortened by Wolf Lammen, 31-Jan-2018) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax13ALT | |- ( -. x = y -> ( y = z -> A. x y = z ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sp | |- ( A. x x = y -> x = y ) |
|
| 2 | 1 | con3i | |- ( -. x = y -> -. A. x x = y ) |
| 3 | sp | |- ( A. x x = z -> x = z ) |
|
| 4 | 3 | con3i | |- ( -. x = z -> -. A. x x = z ) |
| 5 | axc9 | |- ( -. A. x x = y -> ( -. A. x x = z -> ( y = z -> A. x y = z ) ) ) |
|
| 6 | 2 4 5 | syl2im | |- ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) ) |
| 7 | ax13b | |- ( ( -. x = y -> ( y = z -> A. x y = z ) ) <-> ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) ) ) |
|
| 8 | 6 7 | mpbir | |- ( -. x = y -> ( y = z -> A. x y = z ) ) |