Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) |
2 |
1
|
imim2i |
⊢ ( ( 𝑗 ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) → ( 𝑗 ≤ 𝑘 → 𝜑 ) ) |
3 |
2
|
ralimi |
⊢ ( ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) → ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ) |
4 |
3
|
reximi |
⊢ ( ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) → ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ) |
5 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜓 ) |
6 |
5
|
imim2i |
⊢ ( ( 𝑗 ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) → ( 𝑗 ≤ 𝑘 → 𝜓 ) ) |
7 |
6
|
ralimi |
⊢ ( ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) → ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜓 ) ) |
8 |
7
|
reximi |
⊢ ( ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) → ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜓 ) ) |
9 |
4 8
|
jca |
⊢ ( ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) → ( ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ∧ ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜓 ) ) ) |
10 |
|
breq1 |
⊢ ( 𝑗 = 𝑥 → ( 𝑗 ≤ 𝑘 ↔ 𝑥 ≤ 𝑘 ) ) |
11 |
10
|
imbi1d |
⊢ ( 𝑗 = 𝑥 → ( ( 𝑗 ≤ 𝑘 → 𝜑 ) ↔ ( 𝑥 ≤ 𝑘 → 𝜑 ) ) ) |
12 |
11
|
ralbidv |
⊢ ( 𝑗 = 𝑥 → ( ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ↔ ∀ 𝑘 ∈ 𝐴 ( 𝑥 ≤ 𝑘 → 𝜑 ) ) ) |
13 |
12
|
cbvrexvw |
⊢ ( ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑥 ≤ 𝑘 → 𝜑 ) ) |
14 |
|
breq1 |
⊢ ( 𝑗 = 𝑦 → ( 𝑗 ≤ 𝑘 ↔ 𝑦 ≤ 𝑘 ) ) |
15 |
14
|
imbi1d |
⊢ ( 𝑗 = 𝑦 → ( ( 𝑗 ≤ 𝑘 → 𝜓 ) ↔ ( 𝑦 ≤ 𝑘 → 𝜓 ) ) ) |
16 |
15
|
ralbidv |
⊢ ( 𝑗 = 𝑦 → ( ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜓 ) ↔ ∀ 𝑘 ∈ 𝐴 ( 𝑦 ≤ 𝑘 → 𝜓 ) ) ) |
17 |
16
|
cbvrexvw |
⊢ ( ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜓 ) ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑦 ≤ 𝑘 → 𝜓 ) ) |
18 |
13 17
|
anbi12i |
⊢ ( ( ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ∧ ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜓 ) ) ↔ ( ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑥 ≤ 𝑘 → 𝜑 ) ∧ ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑦 ≤ 𝑘 → 𝜓 ) ) ) |
19 |
|
reeanv |
⊢ ( ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ( ∀ 𝑘 ∈ 𝐴 ( 𝑥 ≤ 𝑘 → 𝜑 ) ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑦 ≤ 𝑘 → 𝜓 ) ) ↔ ( ∃ 𝑥 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑥 ≤ 𝑘 → 𝜑 ) ∧ ∃ 𝑦 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑦 ≤ 𝑘 → 𝜓 ) ) ) |
20 |
18 19
|
bitr4i |
⊢ ( ( ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ∧ ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜓 ) ) ↔ ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ( ∀ 𝑘 ∈ 𝐴 ( 𝑥 ≤ 𝑘 → 𝜑 ) ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑦 ≤ 𝑘 → 𝜓 ) ) ) |
21 |
|
ifcl |
⊢ ( ( 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ) → if ( 𝑥 ≤ 𝑦 , 𝑦 , 𝑥 ) ∈ ℝ ) |
22 |
21
|
ancoms |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → if ( 𝑥 ≤ 𝑦 , 𝑦 , 𝑥 ) ∈ ℝ ) |
23 |
22
|
adantl |
⊢ ( ( 𝐴 ⊆ ℝ ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → if ( 𝑥 ≤ 𝑦 , 𝑦 , 𝑥 ) ∈ ℝ ) |
24 |
|
r19.26 |
⊢ ( ∀ 𝑘 ∈ 𝐴 ( ( 𝑥 ≤ 𝑘 → 𝜑 ) ∧ ( 𝑦 ≤ 𝑘 → 𝜓 ) ) ↔ ( ∀ 𝑘 ∈ 𝐴 ( 𝑥 ≤ 𝑘 → 𝜑 ) ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑦 ≤ 𝑘 → 𝜓 ) ) ) |
25 |
|
anim12 |
⊢ ( ( ( 𝑥 ≤ 𝑘 → 𝜑 ) ∧ ( 𝑦 ≤ 𝑘 → 𝜓 ) ) → ( ( 𝑥 ≤ 𝑘 ∧ 𝑦 ≤ 𝑘 ) → ( 𝜑 ∧ 𝜓 ) ) ) |
26 |
|
simplrl |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) ∧ 𝑘 ∈ 𝐴 ) → 𝑥 ∈ ℝ ) |
27 |
|
simplrr |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) ∧ 𝑘 ∈ 𝐴 ) → 𝑦 ∈ ℝ ) |
28 |
|
simpl |
⊢ ( ( 𝐴 ⊆ ℝ ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → 𝐴 ⊆ ℝ ) |
29 |
28
|
sselda |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) ∧ 𝑘 ∈ 𝐴 ) → 𝑘 ∈ ℝ ) |
30 |
|
maxle |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( if ( 𝑥 ≤ 𝑦 , 𝑦 , 𝑥 ) ≤ 𝑘 ↔ ( 𝑥 ≤ 𝑘 ∧ 𝑦 ≤ 𝑘 ) ) ) |
31 |
26 27 29 30
|
syl3anc |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) ∧ 𝑘 ∈ 𝐴 ) → ( if ( 𝑥 ≤ 𝑦 , 𝑦 , 𝑥 ) ≤ 𝑘 ↔ ( 𝑥 ≤ 𝑘 ∧ 𝑦 ≤ 𝑘 ) ) ) |
32 |
31
|
imbi1d |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) ∧ 𝑘 ∈ 𝐴 ) → ( ( if ( 𝑥 ≤ 𝑦 , 𝑦 , 𝑥 ) ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝑥 ≤ 𝑘 ∧ 𝑦 ≤ 𝑘 ) → ( 𝜑 ∧ 𝜓 ) ) ) ) |
33 |
25 32
|
syl5ibr |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) ∧ 𝑘 ∈ 𝐴 ) → ( ( ( 𝑥 ≤ 𝑘 → 𝜑 ) ∧ ( 𝑦 ≤ 𝑘 → 𝜓 ) ) → ( if ( 𝑥 ≤ 𝑦 , 𝑦 , 𝑥 ) ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) ) ) |
34 |
33
|
ralimdva |
⊢ ( ( 𝐴 ⊆ ℝ ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( ∀ 𝑘 ∈ 𝐴 ( ( 𝑥 ≤ 𝑘 → 𝜑 ) ∧ ( 𝑦 ≤ 𝑘 → 𝜓 ) ) → ∀ 𝑘 ∈ 𝐴 ( if ( 𝑥 ≤ 𝑦 , 𝑦 , 𝑥 ) ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) ) ) |
35 |
24 34
|
syl5bir |
⊢ ( ( 𝐴 ⊆ ℝ ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( ( ∀ 𝑘 ∈ 𝐴 ( 𝑥 ≤ 𝑘 → 𝜑 ) ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑦 ≤ 𝑘 → 𝜓 ) ) → ∀ 𝑘 ∈ 𝐴 ( if ( 𝑥 ≤ 𝑦 , 𝑦 , 𝑥 ) ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) ) ) |
36 |
|
breq1 |
⊢ ( 𝑗 = if ( 𝑥 ≤ 𝑦 , 𝑦 , 𝑥 ) → ( 𝑗 ≤ 𝑘 ↔ if ( 𝑥 ≤ 𝑦 , 𝑦 , 𝑥 ) ≤ 𝑘 ) ) |
37 |
36
|
rspceaimv |
⊢ ( ( if ( 𝑥 ≤ 𝑦 , 𝑦 , 𝑥 ) ∈ ℝ ∧ ∀ 𝑘 ∈ 𝐴 ( if ( 𝑥 ≤ 𝑦 , 𝑦 , 𝑥 ) ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) ) → ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) ) |
38 |
23 35 37
|
syl6an |
⊢ ( ( 𝐴 ⊆ ℝ ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( ( ∀ 𝑘 ∈ 𝐴 ( 𝑥 ≤ 𝑘 → 𝜑 ) ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑦 ≤ 𝑘 → 𝜓 ) ) → ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) ) ) |
39 |
38
|
rexlimdvva |
⊢ ( 𝐴 ⊆ ℝ → ( ∃ 𝑥 ∈ ℝ ∃ 𝑦 ∈ ℝ ( ∀ 𝑘 ∈ 𝐴 ( 𝑥 ≤ 𝑘 → 𝜑 ) ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑦 ≤ 𝑘 → 𝜓 ) ) → ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) ) ) |
40 |
20 39
|
syl5bi |
⊢ ( 𝐴 ⊆ ℝ → ( ( ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ∧ ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜓 ) ) → ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) ) ) |
41 |
9 40
|
impbid2 |
⊢ ( 𝐴 ⊆ ℝ → ( ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → ( 𝜑 ∧ 𝜓 ) ) ↔ ( ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜑 ) ∧ ∃ 𝑗 ∈ ℝ ∀ 𝑘 ∈ 𝐴 ( 𝑗 ≤ 𝑘 → 𝜓 ) ) ) ) |