Step |
Hyp |
Ref |
Expression |
1 |
|
breq |
⊢ ( 𝑅 = 𝑆 → ( 𝑧 𝑅 𝑦 ↔ 𝑧 𝑆 𝑦 ) ) |
2 |
1
|
notbid |
⊢ ( 𝑅 = 𝑆 → ( ¬ 𝑧 𝑅 𝑦 ↔ ¬ 𝑧 𝑆 𝑦 ) ) |
3 |
2
|
rexralbidv |
⊢ ( 𝑅 = 𝑆 → ( ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ↔ ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑆 𝑦 ) ) |
4 |
3
|
imbi2d |
⊢ ( 𝑅 = 𝑆 → ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) ↔ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑆 𝑦 ) ) ) |
5 |
4
|
albidv |
⊢ ( 𝑅 = 𝑆 → ( ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) ↔ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑆 𝑦 ) ) ) |
6 |
|
df-fr |
⊢ ( 𝑅 Fr 𝐴 ↔ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) ) |
7 |
|
df-fr |
⊢ ( 𝑆 Fr 𝐴 ↔ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑆 𝑦 ) ) |
8 |
5 6 7
|
3bitr4g |
⊢ ( 𝑅 = 𝑆 → ( 𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴 ) ) |