Step |
Hyp |
Ref |
Expression |
1 |
|
frege129d.f |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
2 |
|
frege129d.a |
⊢ ( 𝜑 → 𝐴 ∈ dom 𝐹 ) |
3 |
|
frege129d.c |
⊢ ( 𝜑 → 𝐶 = ( 𝐹 ‘ 𝐴 ) ) |
4 |
|
frege129d.or |
⊢ ( 𝜑 → ( 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ( t+ ‘ 𝐹 ) 𝐴 ) ) |
5 |
|
frege129d.fun |
⊢ ( 𝜑 → Fun 𝐹 ) |
6 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) → 𝐹 ∈ V ) |
7 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) → 𝐴 ∈ dom 𝐹 ) |
8 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) → 𝐶 = ( 𝐹 ‘ 𝐴 ) ) |
9 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) → 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) |
10 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) → Fun 𝐹 ) |
11 |
6 7 8 9 10
|
frege126d |
⊢ ( ( 𝜑 ∧ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) → ( 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ∨ 𝐶 = 𝐵 ∨ 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ) ) |
12 |
|
biid |
⊢ ( 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ↔ 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ) |
13 |
|
eqcom |
⊢ ( 𝐶 = 𝐵 ↔ 𝐵 = 𝐶 ) |
14 |
|
biid |
⊢ ( 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ↔ 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ) |
15 |
12 13 14
|
3orbi123i |
⊢ ( ( 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ∨ 𝐶 = 𝐵 ∨ 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ) ↔ ( 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ∨ 𝐵 = 𝐶 ∨ 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ) ) |
16 |
11 15
|
sylib |
⊢ ( ( 𝜑 ∧ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) → ( 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ∨ 𝐵 = 𝐶 ∨ 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ) ) |
17 |
|
3orcomb |
⊢ ( ( 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ∨ 𝐵 = 𝐶 ∨ 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ) ↔ ( 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ∨ 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ∨ 𝐵 = 𝐶 ) ) |
18 |
|
3orrot |
⊢ ( ( 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ∨ 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ∨ 𝐵 = 𝐶 ) ↔ ( 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
19 |
17 18
|
sylbb |
⊢ ( ( 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ∨ 𝐵 = 𝐶 ∨ 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ) → ( 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
20 |
16 19
|
syl |
⊢ ( ( 𝜑 ∧ 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ) → ( 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
21 |
20
|
ex |
⊢ ( 𝜑 → ( 𝐴 ( t+ ‘ 𝐹 ) 𝐵 → ( 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ) ) ) |
22 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐴 = 𝐵 ) |
23 |
3
|
eqcomd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = 𝐶 ) |
24 |
|
funbrfvb |
⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝐴 ) = 𝐶 ↔ 𝐴 𝐹 𝐶 ) ) |
25 |
24
|
biimpd |
⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝐴 ) = 𝐶 → 𝐴 𝐹 𝐶 ) ) |
26 |
5 2 25
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐴 ) = 𝐶 → 𝐴 𝐹 𝐶 ) ) |
27 |
23 26
|
mpd |
⊢ ( 𝜑 → 𝐴 𝐹 𝐶 ) |
28 |
1 27
|
frege91d |
⊢ ( 𝜑 → 𝐴 ( t+ ‘ 𝐹 ) 𝐶 ) |
29 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐴 ( t+ ‘ 𝐹 ) 𝐶 ) |
30 |
22 29
|
eqbrtrrd |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ) |
31 |
30
|
ex |
⊢ ( 𝜑 → ( 𝐴 = 𝐵 → 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ) ) |
32 |
|
3mix1 |
⊢ ( 𝐵 ( t+ ‘ 𝐹 ) 𝐶 → ( 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ) ) |
33 |
31 32
|
syl6 |
⊢ ( 𝜑 → ( 𝐴 = 𝐵 → ( 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ) ) ) |
34 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ( t+ ‘ 𝐹 ) 𝐴 ) → 𝐹 ∈ V ) |
35 |
|
funrel |
⊢ ( Fun 𝐹 → Rel 𝐹 ) |
36 |
5 35
|
syl |
⊢ ( 𝜑 → Rel 𝐹 ) |
37 |
|
reltrclfv |
⊢ ( ( 𝐹 ∈ V ∧ Rel 𝐹 ) → Rel ( t+ ‘ 𝐹 ) ) |
38 |
1 36 37
|
syl2anc |
⊢ ( 𝜑 → Rel ( t+ ‘ 𝐹 ) ) |
39 |
|
brrelex1 |
⊢ ( ( Rel ( t+ ‘ 𝐹 ) ∧ 𝐵 ( t+ ‘ 𝐹 ) 𝐴 ) → 𝐵 ∈ V ) |
40 |
38 39
|
sylan |
⊢ ( ( 𝜑 ∧ 𝐵 ( t+ ‘ 𝐹 ) 𝐴 ) → 𝐵 ∈ V ) |
41 |
|
fvex |
⊢ ( 𝐹 ‘ 𝐴 ) ∈ V |
42 |
3 41
|
eqeltrdi |
⊢ ( 𝜑 → 𝐶 ∈ V ) |
43 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ( t+ ‘ 𝐹 ) 𝐴 ) → 𝐶 ∈ V ) |
44 |
2
|
elexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
45 |
44
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ( t+ ‘ 𝐹 ) 𝐴 ) → 𝐴 ∈ V ) |
46 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐵 ( t+ ‘ 𝐹 ) 𝐴 ) → 𝐵 ( t+ ‘ 𝐹 ) 𝐴 ) |
47 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ( t+ ‘ 𝐹 ) 𝐴 ) → 𝐴 𝐹 𝐶 ) |
48 |
34 40 43 45 46 47
|
frege96d |
⊢ ( ( 𝜑 ∧ 𝐵 ( t+ ‘ 𝐹 ) 𝐴 ) → 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ) |
49 |
48
|
ex |
⊢ ( 𝜑 → ( 𝐵 ( t+ ‘ 𝐹 ) 𝐴 → 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ) ) |
50 |
49 32
|
syl6 |
⊢ ( 𝜑 → ( 𝐵 ( t+ ‘ 𝐹 ) 𝐴 → ( 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ) ) ) |
51 |
21 33 50
|
3jaod |
⊢ ( 𝜑 → ( ( 𝐴 ( t+ ‘ 𝐹 ) 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ( t+ ‘ 𝐹 ) 𝐴 ) → ( 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ) ) ) |
52 |
4 51
|
mpd |
⊢ ( 𝜑 → ( 𝐵 ( t+ ‘ 𝐹 ) 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ( t+ ‘ 𝐹 ) 𝐵 ) ) |