Step |
Hyp |
Ref |
Expression |
1 |
|
cbval2v.1 |
⊢ Ⅎ 𝑧 𝜑 |
2 |
|
cbval2v.2 |
⊢ Ⅎ 𝑤 𝜑 |
3 |
|
cbval2v.3 |
⊢ Ⅎ 𝑥 𝜓 |
4 |
|
cbval2v.4 |
⊢ Ⅎ 𝑦 𝜓 |
5 |
|
cbval2v.5 |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) ) |
6 |
1
|
nfal |
⊢ Ⅎ 𝑧 ∀ 𝑦 𝜑 |
7 |
3
|
nfal |
⊢ Ⅎ 𝑥 ∀ 𝑤 𝜓 |
8 |
|
nfv |
⊢ Ⅎ 𝑤 𝑥 = 𝑧 |
9 |
8 2
|
nfim |
⊢ Ⅎ 𝑤 ( 𝑥 = 𝑧 → 𝜑 ) |
10 |
|
nfv |
⊢ Ⅎ 𝑦 𝑥 = 𝑧 |
11 |
10 4
|
nfim |
⊢ Ⅎ 𝑦 ( 𝑥 = 𝑧 → 𝜓 ) |
12 |
5
|
expcom |
⊢ ( 𝑦 = 𝑤 → ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) ) ) |
13 |
12
|
pm5.74d |
⊢ ( 𝑦 = 𝑤 → ( ( 𝑥 = 𝑧 → 𝜑 ) ↔ ( 𝑥 = 𝑧 → 𝜓 ) ) ) |
14 |
9 11 13
|
cbvalv1 |
⊢ ( ∀ 𝑦 ( 𝑥 = 𝑧 → 𝜑 ) ↔ ∀ 𝑤 ( 𝑥 = 𝑧 → 𝜓 ) ) |
15 |
|
19.21v |
⊢ ( ∀ 𝑦 ( 𝑥 = 𝑧 → 𝜑 ) ↔ ( 𝑥 = 𝑧 → ∀ 𝑦 𝜑 ) ) |
16 |
|
19.21v |
⊢ ( ∀ 𝑤 ( 𝑥 = 𝑧 → 𝜓 ) ↔ ( 𝑥 = 𝑧 → ∀ 𝑤 𝜓 ) ) |
17 |
14 15 16
|
3bitr3i |
⊢ ( ( 𝑥 = 𝑧 → ∀ 𝑦 𝜑 ) ↔ ( 𝑥 = 𝑧 → ∀ 𝑤 𝜓 ) ) |
18 |
17
|
pm5.74ri |
⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑦 𝜑 ↔ ∀ 𝑤 𝜓 ) ) |
19 |
6 7 18
|
cbvalv1 |
⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 ↔ ∀ 𝑧 ∀ 𝑤 𝜓 ) |