Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
⊢ ( ( ( 𝑥 = 𝑦 → 𝜑 ) ∧ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) → ( 𝑥 = 𝑦 → 𝜑 ) ) |
2 |
|
pm2.27 |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 = 𝑦 → 𝜑 ) → 𝜑 ) ) |
3 |
2
|
anc2li |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) |
4 |
3
|
sps |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) |
5 |
|
olc |
⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ∨ ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) |
6 |
1 4 5
|
syl56 |
⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ( ( 𝑥 = 𝑦 → 𝜑 ) ∧ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ∨ ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) ) |
7 |
|
simpr |
⊢ ( ( ( 𝑥 = 𝑦 → 𝜑 ) ∧ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) |
8 |
|
equs5 |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
9 |
8
|
biimpd |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
10 |
|
orc |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ∨ ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) |
11 |
7 9 10
|
syl56 |
⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ( ( 𝑥 = 𝑦 → 𝜑 ) ∧ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ∨ ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) ) |
12 |
6 11
|
pm2.61i |
⊢ ( ( ( 𝑥 = 𝑦 → 𝜑 ) ∧ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ∨ ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) |
13 |
|
sp |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑥 = 𝑦 → 𝜑 ) ) |
14 |
|
pm3.4 |
⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → ( 𝑥 = 𝑦 → 𝜑 ) ) |
15 |
13 14
|
jaoi |
⊢ ( ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ∨ ( 𝑥 = 𝑦 ∧ 𝜑 ) ) → ( 𝑥 = 𝑦 → 𝜑 ) ) |
16 |
|
equs4 |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) |
17 |
|
19.8a |
⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) |
18 |
16 17
|
jaoi |
⊢ ( ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ∨ ( 𝑥 = 𝑦 ∧ 𝜑 ) ) → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) |
19 |
15 18
|
jca |
⊢ ( ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ∨ ( 𝑥 = 𝑦 ∧ 𝜑 ) ) → ( ( 𝑥 = 𝑦 → 𝜑 ) ∧ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) |
20 |
12 19
|
impbii |
⊢ ( ( ( 𝑥 = 𝑦 → 𝜑 ) ∧ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ↔ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ∨ ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ) |