\or\imp右插入\and
theorem or_imp_insr_and (A B C: wff): $ (A \or B \imp C) \imp (A \imp C) \and (B \imp C) $;
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | and_comp | (A \imp C) \imp (B \imp C) \imp (A \imp C) \and (B \imp C) |
|
| 2 | 1 | _hyp_null_complete | (A \or B \imp C) \imp (A \imp C) \imp (B \imp C) \imp (A \imp C) \and (B \imp C) |
| 3 | imp_tran | (A \imp A \or B) \imp (A \or B \imp C) \imp A \imp C |
|
| 4 | 3 | _hyp_null_complete | (A \or B \imp C) \imp (A \imp A \or B) \imp (A \or B \imp C) \imp A \imp C |
| 5 | intror_or | A \imp A \or B |
|
| 6 | 5 | _hyp_null_complete | (A \or B \imp C) \imp A \imp A \or B |
| 7 | 4, 6 | _hyp_mp | (A \or B \imp C) \imp (A \or B \imp C) \imp A \imp C |
| 8 | _hyp_null_intro | (A \or B \imp C) \imp A \or B \imp C |
|
| 9 | 7, 8 | _hyp_mp | (A \or B \imp C) \imp A \imp C |
| 10 | 2, 9 | _hyp_mp | (A \or B \imp C) \imp (B \imp C) \imp (A \imp C) \and (B \imp C) |
| 11 | imp_tran | (B \imp A \or B) \imp (A \or B \imp C) \imp B \imp C |
|
| 12 | 11 | _hyp_null_complete | (A \or B \imp C) \imp (B \imp A \or B) \imp (A \or B \imp C) \imp B \imp C |
| 13 | introl_or | B \imp A \or B |
|
| 14 | 13 | _hyp_null_complete | (A \or B \imp C) \imp B \imp A \or B |
| 15 | 12, 14 | _hyp_mp | (A \or B \imp C) \imp (A \or B \imp C) \imp B \imp C |
| 16 | 15, 8 | _hyp_mp | (A \or B \imp C) \imp B \imp C |
| 17 | 10, 16 | _hyp_mp | (A \or B \imp C) \imp (A \imp C) \and (B \imp C) |