Description: Show that the original axiom ax-c4 can be derived from ax-4 ( alim ), ax-10 ( hbn1 ), sp and propositional calculus. See ax4fromc4 for the rederivation of ax-4 from ax-c4 .
Part of the proof is based on the proof of Lemma 22 of Monk2 p. 114. (Contributed by NM, 21-May-2008) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | axc4 | |- ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sp | |- ( A. x -. A. x ph -> -. A. x ph ) |
|
| 2 | 1 | con2i | |- ( A. x ph -> -. A. x -. A. x ph ) |
| 3 | hbn1 | |- ( -. A. x -. A. x ph -> A. x -. A. x -. A. x ph ) |
|
| 4 | hbn1 | |- ( -. A. x ph -> A. x -. A. x ph ) |
|
| 5 | 4 | con1i | |- ( -. A. x -. A. x ph -> A. x ph ) |
| 6 | 5 | alimi | |- ( A. x -. A. x -. A. x ph -> A. x A. x ph ) |
| 7 | 2 3 6 | 3syl | |- ( A. x ph -> A. x A. x ph ) |
| 8 | alim | |- ( A. x ( A. x ph -> ps ) -> ( A. x A. x ph -> A. x ps ) ) |
|
| 9 | 7 8 | syl5 | |- ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) ) |