\iff对称
theorem iff_symm (A B: wff): $ (A \iff B) \imp (B \iff A) $;
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iff_comp | (B \imp A) \imp (A \imp B) \imp (B \iff A) |
|
| 2 | 1 | _hyp_null_complete | (A \iff B) \imp (B \imp A) \imp (A \imp B) \imp (B \iff A) |
| 3 | iff_decompbwd | (A \iff B) \imp B \imp A |
|
| 4 | 3 | _hyp_null_complete | (A \iff B) \imp (A \iff B) \imp B \imp A |
| 5 | _hyp_null_intro | (A \iff B) \imp (A \iff B) |
|
| 6 | 4, 5 | _hyp_mp | (A \iff B) \imp B \imp A |
| 7 | 2, 6 | _hyp_mp | (A \iff B) \imp (A \imp B) \imp (B \iff A) |
| 8 | iff_decomp | (A \iff B) \imp A \imp B |
|
| 9 | 8 | _hyp_null_complete | (A \iff B) \imp (A \iff B) \imp A \imp B |
| 10 | 9, 5 | _hyp_mp | (A \iff B) \imp A \imp B |
| 11 | 7, 10 | _hyp_mp | (A \iff B) \imp (B \iff A) |