Step |
Hyp |
Ref |
Expression |
1 |
|
df-bj-nnf |
⊢ ( Ⅎ' 𝑥 𝜑 ↔ ( ( ∃ 𝑥 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑥 𝜑 ) ) ) |
2 |
|
df-bj-nnf |
⊢ ( Ⅎ' 𝑥 𝜓 ↔ ( ( ∃ 𝑥 𝜓 → 𝜓 ) ∧ ( 𝜓 → ∀ 𝑥 𝜓 ) ) ) |
3 |
|
19.40 |
⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) → ( ∃ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) ) |
4 |
|
anim12 |
⊢ ( ( ( ∃ 𝑥 𝜑 → 𝜑 ) ∧ ( ∃ 𝑥 𝜓 → 𝜓 ) ) → ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) ) |
5 |
3 4
|
syl5 |
⊢ ( ( ( ∃ 𝑥 𝜑 → 𝜑 ) ∧ ( ∃ 𝑥 𝜓 → 𝜓 ) ) → ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) ) |
6 |
|
anim12 |
⊢ ( ( ( 𝜑 → ∀ 𝑥 𝜑 ) ∧ ( 𝜓 → ∀ 𝑥 𝜓 ) ) → ( ( 𝜑 ∧ 𝜓 ) → ( ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) ) ) |
7 |
|
id |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) |
8 |
7
|
alanimi |
⊢ ( ( ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ) |
9 |
6 8
|
syl6 |
⊢ ( ( ( 𝜑 → ∀ 𝑥 𝜑 ) ∧ ( 𝜓 → ∀ 𝑥 𝜓 ) ) → ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) |
10 |
5 9
|
anim12i |
⊢ ( ( ( ( ∃ 𝑥 𝜑 → 𝜑 ) ∧ ( ∃ 𝑥 𝜓 → 𝜓 ) ) ∧ ( ( 𝜑 → ∀ 𝑥 𝜑 ) ∧ ( 𝜓 → ∀ 𝑥 𝜓 ) ) ) → ( ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) ∧ ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) ) |
11 |
10
|
an4s |
⊢ ( ( ( ( ∃ 𝑥 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑥 𝜑 ) ) ∧ ( ( ∃ 𝑥 𝜓 → 𝜓 ) ∧ ( 𝜓 → ∀ 𝑥 𝜓 ) ) ) → ( ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) ∧ ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) ) |
12 |
1 2 11
|
syl2anb |
⊢ ( ( Ⅎ' 𝑥 𝜑 ∧ Ⅎ' 𝑥 𝜓 ) → ( ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) ∧ ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) ) |
13 |
|
df-bj-nnf |
⊢ ( Ⅎ' 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜓 ) ) ∧ ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) ) |
14 |
12 13
|
sylibr |
⊢ ( ( Ⅎ' 𝑥 𝜑 ∧ Ⅎ' 𝑥 𝜓 ) → Ⅎ' 𝑥 ( 𝜑 ∧ 𝜓 ) ) |