Step |
Hyp |
Ref |
Expression |
1 |
|
ltord.1 |
⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 ) |
2 |
|
ltord.2 |
⊢ ( 𝑥 = 𝐶 → 𝐴 = 𝑀 ) |
3 |
|
ltord.3 |
⊢ ( 𝑥 = 𝐷 → 𝐴 = 𝑁 ) |
4 |
|
ltord.4 |
⊢ 𝑆 ⊆ ℝ |
5 |
|
ltord.5 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → 𝐴 ∈ ℝ ) |
6 |
|
ltord.6 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 < 𝑦 → 𝐴 < 𝐵 ) ) |
7 |
6
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 < 𝑦 → 𝐴 < 𝐵 ) ) |
8 |
|
breq1 |
⊢ ( 𝑥 = 𝐶 → ( 𝑥 < 𝑦 ↔ 𝐶 < 𝑦 ) ) |
9 |
2
|
breq1d |
⊢ ( 𝑥 = 𝐶 → ( 𝐴 < 𝐵 ↔ 𝑀 < 𝐵 ) ) |
10 |
8 9
|
imbi12d |
⊢ ( 𝑥 = 𝐶 → ( ( 𝑥 < 𝑦 → 𝐴 < 𝐵 ) ↔ ( 𝐶 < 𝑦 → 𝑀 < 𝐵 ) ) ) |
11 |
|
breq2 |
⊢ ( 𝑦 = 𝐷 → ( 𝐶 < 𝑦 ↔ 𝐶 < 𝐷 ) ) |
12 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝐷 ↔ 𝑦 = 𝐷 ) ) |
13 |
1
|
eqeq1d |
⊢ ( 𝑥 = 𝑦 → ( 𝐴 = 𝑁 ↔ 𝐵 = 𝑁 ) ) |
14 |
12 13
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 = 𝐷 → 𝐴 = 𝑁 ) ↔ ( 𝑦 = 𝐷 → 𝐵 = 𝑁 ) ) ) |
15 |
14 3
|
chvarvv |
⊢ ( 𝑦 = 𝐷 → 𝐵 = 𝑁 ) |
16 |
15
|
breq2d |
⊢ ( 𝑦 = 𝐷 → ( 𝑀 < 𝐵 ↔ 𝑀 < 𝑁 ) ) |
17 |
11 16
|
imbi12d |
⊢ ( 𝑦 = 𝐷 → ( ( 𝐶 < 𝑦 → 𝑀 < 𝐵 ) ↔ ( 𝐶 < 𝐷 → 𝑀 < 𝑁 ) ) ) |
18 |
10 17
|
rspc2v |
⊢ ( ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆 ) → ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 < 𝑦 → 𝐴 < 𝐵 ) → ( 𝐶 < 𝐷 → 𝑀 < 𝑁 ) ) ) |
19 |
7 18
|
mpan9 |
⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆 ) ) → ( 𝐶 < 𝐷 → 𝑀 < 𝑁 ) ) |