Step |
Hyp |
Ref |
Expression |
1 |
|
uun2221p2.1 |
⊢ ( ( ( 𝜓 ∧ 𝜑 ) ∧ 𝜑 ∧ 𝜑 ) → 𝜒 ) |
2 |
|
3anrev |
⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( ( 𝜓 ∧ 𝜑 ) ∧ 𝜑 ∧ 𝜑 ) ) |
3 |
2
|
imbi1i |
⊢ ( ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) → 𝜒 ) ↔ ( ( ( 𝜓 ∧ 𝜑 ) ∧ 𝜑 ∧ 𝜑 ) → 𝜒 ) ) |
4 |
1 3
|
mpbir |
⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) → 𝜒 ) |
5 |
|
3anass |
⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜑 ∧ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) ) |
6 |
|
anabs5 |
⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) |
7 |
5 6
|
bitri |
⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) |
8 |
|
ancom |
⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜑 ) ) |
9 |
8
|
anbi2i |
⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) |
10 |
7 9
|
bitr4i |
⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ) |
11 |
|
anabs5 |
⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ 𝜓 ) ) |
12 |
11 8
|
bitri |
⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜓 ∧ 𝜑 ) ) |
13 |
10 12
|
bitri |
⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜓 ∧ 𝜑 ) ) |
14 |
13
|
imbi1i |
⊢ ( ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) → 𝜒 ) ↔ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 ) ) |
15 |
4 14
|
mpbi |
⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 ) |