Step |
Hyp |
Ref |
Expression |
1 |
|
sbcex |
⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 → 𝐴 ∈ V ) |
2 |
|
sbcex |
⊢ ( [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝜑 → 𝐵 ∈ V ) |
3 |
|
sbc6g |
⊢ ( 𝐵 ∈ V → ( [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝐵 → [ 𝐴 / 𝑥 ] 𝜑 ) ) ) |
4 |
|
isset |
⊢ ( 𝐵 ∈ V ↔ ∃ 𝑦 𝑦 = 𝐵 ) |
5 |
|
exim |
⊢ ( ∀ 𝑦 ( 𝑦 = 𝐵 → [ 𝐴 / 𝑥 ] 𝜑 ) → ( ∃ 𝑦 𝑦 = 𝐵 → ∃ 𝑦 [ 𝐴 / 𝑥 ] 𝜑 ) ) |
6 |
4 5
|
biimtrid |
⊢ ( ∀ 𝑦 ( 𝑦 = 𝐵 → [ 𝐴 / 𝑥 ] 𝜑 ) → ( 𝐵 ∈ V → ∃ 𝑦 [ 𝐴 / 𝑥 ] 𝜑 ) ) |
7 |
|
sbcex |
⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → 𝐴 ∈ V ) |
8 |
7
|
exlimiv |
⊢ ( ∃ 𝑦 [ 𝐴 / 𝑥 ] 𝜑 → 𝐴 ∈ V ) |
9 |
6 8
|
syl6com |
⊢ ( 𝐵 ∈ V → ( ∀ 𝑦 ( 𝑦 = 𝐵 → [ 𝐴 / 𝑥 ] 𝜑 ) → 𝐴 ∈ V ) ) |
10 |
3 9
|
sylbid |
⊢ ( 𝐵 ∈ V → ( [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝜑 → 𝐴 ∈ V ) ) |
11 |
2 10
|
mpcom |
⊢ ( [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝜑 → 𝐴 ∈ V ) |
12 |
1 11
|
pm5.21ni |
⊢ ( ¬ 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝜑 ) ) |
13 |
|
sbc6g |
⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → [ 𝐵 / 𝑦 ] 𝜑 ) ) ) |
14 |
|
isset |
⊢ ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 ) |
15 |
|
exim |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → [ 𝐵 / 𝑦 ] 𝜑 ) → ( ∃ 𝑥 𝑥 = 𝐴 → ∃ 𝑥 [ 𝐵 / 𝑦 ] 𝜑 ) ) |
16 |
14 15
|
biimtrid |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → [ 𝐵 / 𝑦 ] 𝜑 ) → ( 𝐴 ∈ V → ∃ 𝑥 [ 𝐵 / 𝑦 ] 𝜑 ) ) |
17 |
|
sbcex |
⊢ ( [ 𝐵 / 𝑦 ] 𝜑 → 𝐵 ∈ V ) |
18 |
17
|
exlimiv |
⊢ ( ∃ 𝑥 [ 𝐵 / 𝑦 ] 𝜑 → 𝐵 ∈ V ) |
19 |
16 18
|
syl6com |
⊢ ( 𝐴 ∈ V → ( ∀ 𝑥 ( 𝑥 = 𝐴 → [ 𝐵 / 𝑦 ] 𝜑 ) → 𝐵 ∈ V ) ) |
20 |
13 19
|
sylbid |
⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 → 𝐵 ∈ V ) ) |
21 |
1 20
|
mpcom |
⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 → 𝐵 ∈ V ) |
22 |
21 2
|
pm5.21ni |
⊢ ( ¬ 𝐵 ∈ V → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝜑 ) ) |
23 |
|
bi2.04 |
⊢ ( ( 𝑥 = 𝐴 → ( 𝑦 = 𝐵 → 𝜑 ) ) ↔ ( 𝑦 = 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
24 |
23
|
2albii |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 = 𝐴 → ( 𝑦 = 𝐵 → 𝜑 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ( 𝑦 = 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
25 |
|
alcom |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑦 = 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ∀ 𝑦 ∀ 𝑥 ( 𝑦 = 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
26 |
24 25
|
bitri |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 = 𝐴 → ( 𝑦 = 𝐵 → 𝜑 ) ) ↔ ∀ 𝑦 ∀ 𝑥 ( 𝑦 = 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
27 |
|
19.21v |
⊢ ( ∀ 𝑦 ( 𝑥 = 𝐴 → ( 𝑦 = 𝐵 → 𝜑 ) ) ↔ ( 𝑥 = 𝐴 → ∀ 𝑦 ( 𝑦 = 𝐵 → 𝜑 ) ) ) |
28 |
27
|
albii |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 = 𝐴 → ( 𝑦 = 𝐵 → 𝜑 ) ) ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → ∀ 𝑦 ( 𝑦 = 𝐵 → 𝜑 ) ) ) |
29 |
|
19.21v |
⊢ ( ∀ 𝑥 ( 𝑦 = 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ( 𝑦 = 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
30 |
29
|
albii |
⊢ ( ∀ 𝑦 ∀ 𝑥 ( 𝑦 = 𝐵 → ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ∀ 𝑦 ( 𝑦 = 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
31 |
26 28 30
|
3bitr3i |
⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ∀ 𝑦 ( 𝑦 = 𝐵 → 𝜑 ) ) ↔ ∀ 𝑦 ( 𝑦 = 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
32 |
31
|
a1i |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → ∀ 𝑦 ( 𝑦 = 𝐵 → 𝜑 ) ) ↔ ∀ 𝑦 ( 𝑦 = 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) ) |
33 |
|
sbc6g |
⊢ ( 𝐵 ∈ V → ( [ 𝐵 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝐵 → 𝜑 ) ) ) |
34 |
33
|
imbi2d |
⊢ ( 𝐵 ∈ V → ( ( 𝑥 = 𝐴 → [ 𝐵 / 𝑦 ] 𝜑 ) ↔ ( 𝑥 = 𝐴 → ∀ 𝑦 ( 𝑦 = 𝐵 → 𝜑 ) ) ) ) |
35 |
34
|
albidv |
⊢ ( 𝐵 ∈ V → ( ∀ 𝑥 ( 𝑥 = 𝐴 → [ 𝐵 / 𝑦 ] 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → ∀ 𝑦 ( 𝑦 = 𝐵 → 𝜑 ) ) ) ) |
36 |
35
|
adantl |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → [ 𝐵 / 𝑦 ] 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → ∀ 𝑦 ( 𝑦 = 𝐵 → 𝜑 ) ) ) ) |
37 |
|
sbc6g |
⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) |
38 |
37
|
imbi2d |
⊢ ( 𝐴 ∈ V → ( ( 𝑦 = 𝐵 → [ 𝐴 / 𝑥 ] 𝜑 ) ↔ ( 𝑦 = 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) ) |
39 |
38
|
albidv |
⊢ ( 𝐴 ∈ V → ( ∀ 𝑦 ( 𝑦 = 𝐵 → [ 𝐴 / 𝑥 ] 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 = 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) ) |
40 |
39
|
adantr |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ∀ 𝑦 ( 𝑦 = 𝐵 → [ 𝐴 / 𝑥 ] 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 = 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) ) |
41 |
32 36 40
|
3bitr4d |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → [ 𝐵 / 𝑦 ] 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 = 𝐵 → [ 𝐴 / 𝑥 ] 𝜑 ) ) ) |
42 |
13
|
adantr |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → [ 𝐵 / 𝑦 ] 𝜑 ) ) ) |
43 |
3
|
adantl |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝐵 → [ 𝐴 / 𝑥 ] 𝜑 ) ) ) |
44 |
41 42 43
|
3bitr4d |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝜑 ) ) |
45 |
12 22 44
|
ecase |
⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝜑 ) |