Step |
Hyp |
Ref |
Expression |
1 |
|
cgsex4gOLD.1 |
⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → 𝜒 ) |
2 |
|
cgsex4gOLD.2 |
⊢ ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) |
3 |
2
|
biimpa |
⊢ ( ( 𝜒 ∧ 𝜑 ) → 𝜓 ) |
4 |
3
|
exlimivv |
⊢ ( ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) → 𝜓 ) |
5 |
4
|
exlimivv |
⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) → 𝜓 ) |
6 |
|
elisset |
⊢ ( 𝐴 ∈ 𝑅 → ∃ 𝑥 𝑥 = 𝐴 ) |
7 |
|
elisset |
⊢ ( 𝐵 ∈ 𝑆 → ∃ 𝑦 𝑦 = 𝐵 ) |
8 |
6 7
|
anim12i |
⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) ) |
9 |
|
exdistrv |
⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) ) |
10 |
8 9
|
sylibr |
⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) |
11 |
|
elisset |
⊢ ( 𝐶 ∈ 𝑅 → ∃ 𝑧 𝑧 = 𝐶 ) |
12 |
|
elisset |
⊢ ( 𝐷 ∈ 𝑆 → ∃ 𝑤 𝑤 = 𝐷 ) |
13 |
11 12
|
anim12i |
⊢ ( ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) → ( ∃ 𝑧 𝑧 = 𝐶 ∧ ∃ 𝑤 𝑤 = 𝐷 ) ) |
14 |
|
exdistrv |
⊢ ( ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ↔ ( ∃ 𝑧 𝑧 = 𝐶 ∧ ∃ 𝑤 𝑤 = 𝐷 ) ) |
15 |
13 14
|
sylibr |
⊢ ( ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) |
16 |
10 15
|
anim12i |
⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) |
17 |
|
eqeq1 |
⊢ ( 𝑦 = 𝑣 → ( 𝑦 = 𝐵 ↔ 𝑣 = 𝐵 ) ) |
18 |
17
|
anbi2d |
⊢ ( 𝑦 = 𝑣 → ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( 𝑥 = 𝐴 ∧ 𝑣 = 𝐵 ) ) ) |
19 |
18
|
anbi1d |
⊢ ( 𝑦 = 𝑣 → ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ( 𝑥 = 𝐴 ∧ 𝑣 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) ) |
20 |
19
|
exbidv |
⊢ ( 𝑦 = 𝑣 → ( ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑣 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) ) |
21 |
20
|
notbid |
⊢ ( 𝑦 = 𝑣 → ( ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑣 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) ) |
22 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑣 → ( 𝑧 = 𝐶 ↔ 𝑣 = 𝐶 ) ) |
23 |
22
|
anbi1d |
⊢ ( 𝑧 = 𝑣 → ( ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ↔ ( 𝑣 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) |
24 |
23
|
anbi2d |
⊢ ( 𝑧 = 𝑣 → ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑣 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) ) |
25 |
24
|
exbidv |
⊢ ( 𝑧 = 𝑣 → ( ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑣 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) ) |
26 |
25
|
notbid |
⊢ ( 𝑧 = 𝑣 → ( ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑣 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) ) |
27 |
21 26
|
alcomw |
⊢ ( ∀ 𝑦 ∀ 𝑧 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ∀ 𝑧 ∀ 𝑦 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) |
28 |
27
|
notbii |
⊢ ( ¬ ∀ 𝑦 ∀ 𝑧 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ¬ ∀ 𝑧 ∀ 𝑦 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) |
29 |
|
2exnaln |
⊢ ( ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ¬ ∀ 𝑦 ∀ 𝑧 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) |
30 |
|
2exnaln |
⊢ ( ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ¬ ∀ 𝑧 ∀ 𝑦 ¬ ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) |
31 |
28 29 30
|
3bitr4i |
⊢ ( ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) |
32 |
31
|
exbii |
⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ∃ 𝑥 ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) |
33 |
|
4exdistrv |
⊢ ( ∃ 𝑥 ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) |
34 |
32 33
|
bitri |
⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ↔ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) |
35 |
16 34
|
sylibr |
⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ) |
36 |
1
|
2eximi |
⊢ ( ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → ∃ 𝑧 ∃ 𝑤 𝜒 ) |
37 |
36
|
2eximi |
⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜒 ) |
38 |
35 37
|
syl |
⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜒 ) |
39 |
2
|
biimprcd |
⊢ ( 𝜓 → ( 𝜒 → 𝜑 ) ) |
40 |
39
|
ancld |
⊢ ( 𝜓 → ( 𝜒 → ( 𝜒 ∧ 𝜑 ) ) ) |
41 |
40
|
2eximdv |
⊢ ( 𝜓 → ( ∃ 𝑧 ∃ 𝑤 𝜒 → ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ) ) |
42 |
41
|
2eximdv |
⊢ ( 𝜓 → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 𝜒 → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ) ) |
43 |
38 42
|
syl5com |
⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( 𝜓 → ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ) ) |
44 |
5 43
|
impbid2 |
⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜒 ∧ 𝜑 ) ↔ 𝜓 ) ) |