Step |
Hyp |
Ref |
Expression |
1 |
|
nnindd.1 |
⊢ ( 𝑥 = 1 → ( 𝜓 ↔ 𝜒 ) ) |
2 |
|
nnindd.2 |
⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜃 ) ) |
3 |
|
nnindd.3 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝜓 ↔ 𝜏 ) ) |
4 |
|
nnindd.4 |
⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜂 ) ) |
5 |
|
nnindd.5 |
⊢ ( 𝜑 → 𝜒 ) |
6 |
|
nnindd.6 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ ) ∧ 𝜃 ) → 𝜏 ) |
7 |
1
|
imbi2d |
⊢ ( 𝑥 = 1 → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) ) |
8 |
2
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜃 ) ) ) |
9 |
3
|
imbi2d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜏 ) ) ) |
10 |
4
|
imbi2d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜂 ) ) ) |
11 |
6
|
ex |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℕ ) → ( 𝜃 → 𝜏 ) ) |
12 |
11
|
expcom |
⊢ ( 𝑦 ∈ ℕ → ( 𝜑 → ( 𝜃 → 𝜏 ) ) ) |
13 |
12
|
a2d |
⊢ ( 𝑦 ∈ ℕ → ( ( 𝜑 → 𝜃 ) → ( 𝜑 → 𝜏 ) ) ) |
14 |
7 8 9 10 5 13
|
nnind |
⊢ ( 𝐴 ∈ ℕ → ( 𝜑 → 𝜂 ) ) |
15 |
14
|
impcom |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℕ ) → 𝜂 ) |