Description: Show that ax-c11 can be derived from ax-c11n and ax-12 . An open problem is whether this theorem can be derived from ax-c11n and the others when ax-12 is replaced with ax-c15 or ax12v . See Theorems axc11nfromc11 for the rederivation of ax-c11n from axc11 .
Normally, axc11 should be used rather than ax-c11 or axc11-o , except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | axc11-o | |- ( A. x x = y -> ( A. x ph -> A. y ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-c11n | |- ( A. x x = y -> A. y y = x ) |
|
| 2 | ax12 | |- ( y = x -> ( A. x ph -> A. y ( y = x -> ph ) ) ) |
|
| 3 | 2 | equcoms | |- ( x = y -> ( A. x ph -> A. y ( y = x -> ph ) ) ) |
| 4 | 3 | sps-o | |- ( A. x x = y -> ( A. x ph -> A. y ( y = x -> ph ) ) ) |
| 5 | pm2.27 | |- ( y = x -> ( ( y = x -> ph ) -> ph ) ) |
|
| 6 | 5 | al2imi | |- ( A. y y = x -> ( A. y ( y = x -> ph ) -> A. y ph ) ) |
| 7 | 1 4 6 | sylsyld | |- ( A. x x = y -> ( A. x ph -> A. y ph ) ) |