Description: It is possible to remove any restriction on ph in ax12v . Same as Axiom C8 of Monk2 p. 105. Use ax12v instead when sufficient. (Contributed by NM, 5-Aug-1993) Remove dependencies on ax-10 and ax-13 . (Revised by Jim Kingdon, 15-Dec-2017) (Proof shortened by Wolf Lammen, 8-Dec-2019)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax12v2 | |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equtrr | |- ( y = z -> ( x = y -> x = z ) ) |
|
| 2 | ax12v | |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) ) |
|
| 3 | 1 | imim1d | |- ( y = z -> ( ( x = z -> ph ) -> ( x = y -> ph ) ) ) |
| 4 | 3 | alimdv | |- ( y = z -> ( A. x ( x = z -> ph ) -> A. x ( x = y -> ph ) ) ) |
| 5 | 2 4 | syl9r | |- ( y = z -> ( x = z -> ( ph -> A. x ( x = y -> ph ) ) ) ) |
| 6 | 1 5 | syld | |- ( y = z -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) |
| 7 | ax6evr | |- E. z y = z |
|
| 8 | 6 7 | exlimiiv | |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) |