Step |
Hyp |
Ref |
Expression |
1 |
|
elabd2.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
elabd2.2 |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) |
3 |
|
elab6g |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ { 𝑥 ∣ 𝜓 } ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ) ) |
4 |
3
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ∈ { 𝑥 ∣ 𝜓 } ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ) ) |
5 |
|
elisset |
⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 ) |
6 |
2
|
pm5.74da |
⊢ ( 𝜑 → ( ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( 𝑥 = 𝐴 → 𝜒 ) ) ) |
7 |
6
|
albidv |
⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜒 ) ) ) |
8 |
|
19.23v |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜒 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜒 ) ) |
9 |
7 8
|
bitrdi |
⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜒 ) ) ) |
10 |
|
pm5.5 |
⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( ( ∃ 𝑥 𝑥 = 𝐴 → 𝜒 ) ↔ 𝜒 ) ) |
11 |
9 10
|
sylan9bb |
⊢ ( ( 𝜑 ∧ ∃ 𝑥 𝑥 = 𝐴 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ 𝜒 ) ) |
12 |
5 11
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ 𝜒 ) ) |
13 |
4 12
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ∈ { 𝑥 ∣ 𝜓 } ↔ 𝜒 ) ) |
14 |
1 13
|
mpdan |
⊢ ( 𝜑 → ( 𝐴 ∈ { 𝑥 ∣ 𝜓 } ↔ 𝜒 ) ) |