Description: Third lemma for the derivation of jarr and a commuted form of ax-2 , and indirectly ax-1 and ax-2 proper , from adh-minimp and ax-mp . Polish prefix notation: CCCCpqCCCrpCqsCpstt . (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | adh-minimp-jarr-ax2c-lem3 | |- ( ( ( ( ph -> ps ) -> ( ( ( ch -> ph ) -> ( ps -> th ) ) -> ( ph -> th ) ) ) -> ta ) -> ta ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | adh-minimp-jarr-lem2 | |- ( ( ( et -> ( ( ze -> si ) -> ( ( ( rh -> ze ) -> ( si -> mu ) ) -> ( ze -> mu ) ) ) ) -> ( ( ( ph -> ps ) -> ( ( ( ch -> ph ) -> ( ps -> th ) ) -> ( ph -> th ) ) ) -> ta ) ) -> ( ( ( ze -> si ) -> ( ( ( rh -> ze ) -> ( si -> mu ) ) -> ( ze -> mu ) ) ) -> ta ) ) |
|
| 2 | adh-minimp-jarr-lem2 | |- ( ( ( ( et -> ( ( ze -> si ) -> ( ( ( rh -> ze ) -> ( si -> mu ) ) -> ( ze -> mu ) ) ) ) -> ( ( ( ph -> ps ) -> ( ( ( ch -> ph ) -> ( ps -> th ) ) -> ( ph -> th ) ) ) -> ta ) ) -> ( ( ( ze -> si ) -> ( ( ( rh -> ze ) -> ( si -> mu ) ) -> ( ze -> mu ) ) ) -> ta ) ) -> ( ( ( ( ph -> ps ) -> ( ( ( ch -> ph ) -> ( ps -> th ) ) -> ( ph -> th ) ) ) -> ta ) -> ta ) ) |
|
| 3 | 1 2 | ax-mp | |- ( ( ( ( ph -> ps ) -> ( ( ( ch -> ph ) -> ( ps -> th ) ) -> ( ph -> th ) ) ) -> ta ) -> ta ) |