Description: Alternate proof of ax12v2 , shorter, but depending on more axioms. (Contributed by NM, 5-Aug-1993) (New usage is discouraged.) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax12vALT | |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-1 | |- ( ph -> ( x = y -> ph ) ) |
|
| 2 | axc16 | |- ( A. x x = y -> ( ( x = y -> ph ) -> A. x ( x = y -> ph ) ) ) |
|
| 3 | 1 2 | syl5 | |- ( A. x x = y -> ( ph -> A. x ( x = y -> ph ) ) ) |
| 4 | 3 | a1d | |- ( A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) |
| 5 | axc15 | |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) |
|
| 6 | 4 5 | pm2.61i | |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) |