\iff同时引入\and
theorem iff_simintro_and (A B C D: wff): $ (A \iff B) \imp (C \iff D) \imp (A \and C \iff B \and D) $;
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iff_intro_neg | (A \imp \neg C \iff B \imp \neg D) \imp (\neg (A \imp \neg C) \iff \neg (B \imp \neg D)) |
|
| 2 | 1 | _hyp_null_complete | (A \iff B) \and (C \iff D) \imp (A \imp \neg C \iff B \imp \neg D) \imp (\neg (A \imp \neg C) \iff \neg (B \imp \neg D)) |
| 3 | iff_simintro_imp | (A \iff B) \imp (\neg C \iff \neg D) \imp (A \imp \neg C \iff B \imp \neg D) |
|
| 4 | 3 | _hyp_null_complete | (A \iff B) \and (C \iff D) \imp (A \iff B) \imp (\neg C \iff \neg D) \imp (A \imp \neg C \iff B \imp \neg D) |
| 5 | _hyp_null_intro | (A \iff B) \imp (A \iff B) |
|
| 6 | 5 | _hyp_complete | (A \iff B) \and (C \iff D) \imp (A \iff B) |
| 7 | 4, 6 | _hyp_mp | (A \iff B) \and (C \iff D) \imp (\neg C \iff \neg D) \imp (A \imp \neg C \iff B \imp \neg D) |
| 8 | iff_intro_neg | (C \iff D) \imp (\neg C \iff \neg D) |
|
| 9 | 8 | _hyp_null_complete | (A \iff B) \and (C \iff D) \imp (C \iff D) \imp (\neg C \iff \neg D) |
| 10 | _hyp_intro | (A \iff B) \and (C \iff D) \imp (C \iff D) |
|
| 11 | 9, 10 | _hyp_mp | (A \iff B) \and (C \iff D) \imp (\neg C \iff \neg D) |
| 12 | 7, 11 | _hyp_mp | (A \iff B) \and (C \iff D) \imp (A \imp \neg C \iff B \imp \neg D) |
| 13 | 2, 12 | _hyp_mp | (A \iff B) \and (C \iff D) \imp (\neg (A \imp \neg C) \iff \neg (B \imp \neg D)) |
| 14 | 13 | conv and | (A \iff B) \and (C \iff D) \imp (A \and C \iff B \and D) |
| 15 | 14 | _hyp_rm | (A \iff B) \imp (C \iff D) \imp (A \and C \iff B \and D) |