Step |
Hyp |
Ref |
Expression |
1 |
|
2mos.1 |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) ) |
2 |
|
2mo |
⊢ ( ∃ 𝑧 ∃ 𝑤 ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ) |
3 |
1
|
2sbievw |
⊢ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ 𝜓 ) |
4 |
3
|
anbi2i |
⊢ ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) ↔ ( 𝜑 ∧ 𝜓 ) ) |
5 |
4
|
imbi1i |
⊢ ( ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ) |
6 |
5
|
2albii |
⊢ ( ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ) |
7 |
6
|
2albii |
⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ) |
8 |
2 7
|
bitri |
⊢ ( ∃ 𝑧 ∃ 𝑤 ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ) |