Step |
Hyp |
Ref |
Expression |
1 |
|
con3 |
⊢ ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜑 ) ) |
2 |
1
|
orim2d |
⊢ ( ( 𝜑 → 𝜓 ) → ( ( ¬ 𝜒 ∨ ¬ 𝜓 ) → ( ¬ 𝜒 ∨ ¬ 𝜑 ) ) ) |
3 |
|
pm3.13 |
⊢ ( ¬ ( 𝜒 ∧ 𝜓 ) → ( ¬ 𝜒 ∨ ¬ 𝜓 ) ) |
4 |
|
pm3.14 |
⊢ ( ( ¬ 𝜒 ∨ ¬ 𝜑 ) → ¬ ( 𝜒 ∧ 𝜑 ) ) |
5 |
3 4
|
imim12i |
⊢ ( ( ( ¬ 𝜒 ∨ ¬ 𝜓 ) → ( ¬ 𝜒 ∨ ¬ 𝜑 ) ) → ( ¬ ( 𝜒 ∧ 𝜓 ) → ¬ ( 𝜒 ∧ 𝜑 ) ) ) |
6 |
|
df-nan |
⊢ ( ( 𝜒 ⊼ 𝜓 ) ↔ ¬ ( 𝜒 ∧ 𝜓 ) ) |
7 |
|
df-nan |
⊢ ( ( 𝜒 ⊼ 𝜑 ) ↔ ¬ ( 𝜒 ∧ 𝜑 ) ) |
8 |
5 6 7
|
3imtr4g |
⊢ ( ( ( ¬ 𝜒 ∨ ¬ 𝜓 ) → ( ¬ 𝜒 ∨ ¬ 𝜑 ) ) → ( ( 𝜒 ⊼ 𝜓 ) → ( 𝜒 ⊼ 𝜑 ) ) ) |
9 |
2 8
|
syl |
⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 ⊼ 𝜓 ) → ( 𝜒 ⊼ 𝜑 ) ) ) |