Step |
Hyp |
Ref |
Expression |
1 |
|
uun2221.1 |
⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) → 𝜒 ) |
2 |
|
3anass |
⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜑 ∧ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) ) |
3 |
|
anabs5 |
⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) |
4 |
2 3
|
bitri |
⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) |
5 |
|
ancom |
⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜑 ) ) |
6 |
5
|
anbi2i |
⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) |
7 |
4 6
|
bitr4i |
⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ) |
8 |
|
anabs5 |
⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ 𝜓 ) ) |
9 |
8 5
|
bitri |
⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜓 ∧ 𝜑 ) ) |
10 |
7 9
|
bitri |
⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜓 ∧ 𝜑 ) ) |
11 |
10
|
imbi1i |
⊢ ( ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) → 𝜒 ) ↔ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 ) ) |
12 |
1 11
|
mpbi |
⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 ) |