Step |
Hyp |
Ref |
Expression |
1 |
|
ax6e |
⊢ ∃ 𝑧 𝑧 = 𝑥 |
2 |
|
nfnae |
⊢ Ⅎ 𝑧 ¬ ∀ 𝑤 𝑤 = 𝑧 |
3 |
|
nfnae |
⊢ Ⅎ 𝑧 ¬ ∀ 𝑤 𝑤 = 𝑥 |
4 |
2 3
|
nfan |
⊢ Ⅎ 𝑧 ( ¬ ∀ 𝑤 𝑤 = 𝑧 ∧ ¬ ∀ 𝑤 𝑤 = 𝑥 ) |
5 |
|
nfeqf |
⊢ ( ( ¬ ∀ 𝑤 𝑤 = 𝑧 ∧ ¬ ∀ 𝑤 𝑤 = 𝑥 ) → Ⅎ 𝑤 𝑧 = 𝑥 ) |
6 |
|
pm3.21 |
⊢ ( 𝑤 = 𝑦 → ( 𝑧 = 𝑥 → ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) ) |
7 |
5 6
|
spimed |
⊢ ( ( ¬ ∀ 𝑤 𝑤 = 𝑧 ∧ ¬ ∀ 𝑤 𝑤 = 𝑥 ) → ( 𝑧 = 𝑥 → ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) ) |
8 |
4 7
|
eximd |
⊢ ( ( ¬ ∀ 𝑤 𝑤 = 𝑧 ∧ ¬ ∀ 𝑤 𝑤 = 𝑥 ) → ( ∃ 𝑧 𝑧 = 𝑥 → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) ) |
9 |
1 8
|
mpi |
⊢ ( ( ¬ ∀ 𝑤 𝑤 = 𝑧 ∧ ¬ ∀ 𝑤 𝑤 = 𝑥 ) → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) |
10 |
9
|
ex |
⊢ ( ¬ ∀ 𝑤 𝑤 = 𝑧 → ( ¬ ∀ 𝑤 𝑤 = 𝑥 → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) ) |
11 |
|
ax6e |
⊢ ∃ 𝑧 𝑧 = 𝑦 |
12 |
|
nfae |
⊢ Ⅎ 𝑧 ∀ 𝑤 𝑤 = 𝑥 |
13 |
|
equvini |
⊢ ( 𝑧 = 𝑦 → ∃ 𝑤 ( 𝑧 = 𝑤 ∧ 𝑤 = 𝑦 ) ) |
14 |
|
equtrr |
⊢ ( 𝑤 = 𝑥 → ( 𝑧 = 𝑤 → 𝑧 = 𝑥 ) ) |
15 |
14
|
anim1d |
⊢ ( 𝑤 = 𝑥 → ( ( 𝑧 = 𝑤 ∧ 𝑤 = 𝑦 ) → ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) ) |
16 |
15
|
aleximi |
⊢ ( ∀ 𝑤 𝑤 = 𝑥 → ( ∃ 𝑤 ( 𝑧 = 𝑤 ∧ 𝑤 = 𝑦 ) → ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) ) |
17 |
13 16
|
syl5 |
⊢ ( ∀ 𝑤 𝑤 = 𝑥 → ( 𝑧 = 𝑦 → ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) ) |
18 |
12 17
|
eximd |
⊢ ( ∀ 𝑤 𝑤 = 𝑥 → ( ∃ 𝑧 𝑧 = 𝑦 → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) ) |
19 |
11 18
|
mpi |
⊢ ( ∀ 𝑤 𝑤 = 𝑥 → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) |
20 |
10 19
|
pm2.61d2 |
⊢ ( ¬ ∀ 𝑤 𝑤 = 𝑧 → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) |