\fo\and分配左侧(右侧不出现)
theorem fo_and_distls_nfrs {x: set} (A B: wff x):
$ \nf x B $ >
$ \fo x (A \and B) \iff \fo x A \and B $;
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iff_tran | (\fo x (A \and B) \iff \fo x A \and \fo x B) \imp (\fo x A \and \fo x B \iff \fo x A \and B) \imp (\fo x (A \and B) \iff \fo x A \and B) |
|
| 2 | fo_and_dist | \fo x (A \and B) \iff \fo x A \and \fo x B |
|
| 3 | 1, 2 | ax_mp | (\fo x A \and \fo x B \iff \fo x A \and B) \imp (\fo x (A \and B) \iff \fo x A \and B) |
| 4 | iff_introl_and | (\fo x B \iff B) \imp (\fo x A \and \fo x B \iff \fo x A \and B) |
|
| 5 | iff_symm | (B \iff \fo x B) \imp (\fo x B \iff B) |
|
| 6 | hyp n | \nf x B |
|
| 7 | 6 | fo_alloc_nf | B \iff \fo x B |
| 8 | 5, 7 | ax_mp | \fo x B \iff B |
| 9 | 4, 8 | ax_mp | \fo x A \and \fo x B \iff \fo x A \and B |
| 10 | 3, 9 | ax_mp | \fo x (A \and B) \iff \fo x A \and B |